LMFDB - Number field 28.2.725917312055025700242151478457681268310546875.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625)

gp: K = bnfinit(y^28 - 5*y^27 - 3*y^26 + 57*y^25 - 13*y^24 - 420*y^23 + 259*y^22 + 2099*y^21 - 1147*y^20 - 7975*y^19 + 7844*y^18 + 15881*y^17 - 34395*y^16 - 17074*y^15 + 107194*y^14 - 103829*y^13 + 94772*y^12 - 84342*y^11 - 113950*y^10 + 177636*y^9 - 26899*y^8 + 2410*y^7 + 74025*y^6 - 120275*y^5 - 49775*y^4 + 101875*y^3 - 41875*y^2 + 31250*y - 15625, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625)

$ x^{28} - 5 x^{27} - 3 x^{26} + 57 x^{25} - 13 x^{24} - 420 x^{23} + 259 x^{22} + 2099 x^{21} + \cdots - 15625 $ x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$28$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[2, 13]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-725917312055025700242151478457681268310546875$ -725917312055025700242151478457681268310546875 $\medspace = -\,5^{14}\cdot 499^{13}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$40.01$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$5^{1/2}499^{1/2}\approx 49.949974974968704$
Ramified primes:	$5$, $499$ 5, 499	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-499}) $
$\card{ \Aut(K/\Q) }$:	$2$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{6}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{7}+\frac{1}{5}a^{3}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{25}a^{14}+\frac{2}{25}a^{12}+\frac{2}{25}a^{11}+\frac{2}{25}a^{10}+\frac{1}{25}a^{9}-\frac{1}{25}a^{8}+\frac{1}{25}a^{7}+\frac{1}{5}a^{6}-\frac{3}{25}a^{5}-\frac{3}{25}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{25}a^{15}+\frac{2}{25}a^{13}+\frac{2}{25}a^{12}+\frac{2}{25}a^{11}+\frac{1}{25}a^{10}-\frac{1}{25}a^{9}+\frac{1}{25}a^{8}+\frac{1}{5}a^{7}-\frac{3}{25}a^{6}-\frac{3}{25}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}$, $\frac{1}{25}a^{16}+\frac{2}{25}a^{13}-\frac{2}{25}a^{12}+\frac{2}{25}a^{11}-\frac{1}{25}a^{9}+\frac{2}{25}a^{8}+\frac{12}{25}a^{6}+\frac{6}{25}a^{5}+\frac{6}{25}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{17}-\frac{2}{25}a^{13}-\frac{2}{25}a^{12}+\frac{1}{25}a^{11}+\frac{2}{25}a^{8}+\frac{11}{25}a^{6}+\frac{12}{25}a^{5}+\frac{11}{25}a^{4}+\frac{2}{5}a^{3}$, $\frac{1}{25}a^{18}-\frac{2}{25}a^{13}-\frac{1}{25}a^{11}-\frac{1}{25}a^{10}-\frac{1}{25}a^{9}-\frac{2}{25}a^{8}+\frac{8}{25}a^{7}+\frac{12}{25}a^{6}-\frac{2}{5}a^{5}-\frac{11}{25}a^{4}+\frac{1}{5}a^{3}$, $\frac{1}{125}a^{19}-\frac{2}{125}a^{18}-\frac{1}{125}a^{17}+\frac{2}{125}a^{16}+\frac{1}{125}a^{15}-\frac{1}{125}a^{14}+\frac{7}{125}a^{13}+\frac{6}{125}a^{12}+\frac{8}{125}a^{11}+\frac{9}{125}a^{10}+\frac{3}{125}a^{9}+\frac{9}{125}a^{8}-\frac{8}{125}a^{7}-\frac{29}{125}a^{6}-\frac{7}{125}a^{5}-\frac{3}{25}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{125}a^{20}+\frac{1}{125}a^{15}-\frac{1}{25}a^{12}+\frac{6}{125}a^{10}+\frac{2}{25}a^{9}-\frac{1}{25}a^{8}+\frac{3}{25}a^{7}+\frac{2}{25}a^{6}-\frac{44}{125}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{125}a^{21}+\frac{1}{125}a^{16}-\frac{1}{25}a^{13}+\frac{6}{125}a^{11}+\frac{2}{25}a^{10}-\frac{1}{25}a^{9}-\frac{2}{25}a^{8}-\frac{8}{25}a^{7}+\frac{6}{125}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{2}$, $\frac{1}{625}a^{22}+\frac{2}{625}a^{21}-\frac{1}{625}a^{20}+\frac{2}{625}a^{19}-\frac{9}{625}a^{18}+\frac{4}{625}a^{17}-\frac{9}{625}a^{16}-\frac{4}{625}a^{15}-\frac{2}{625}a^{14}+\frac{39}{625}a^{13}-\frac{32}{625}a^{12}-\frac{57}{625}a^{11}+\frac{62}{625}a^{10}+\frac{56}{625}a^{9}+\frac{18}{625}a^{8}+\frac{2}{125}a^{7}+\frac{99}{625}a^{6}+\frac{1}{5}a^{5}+\frac{11}{25}a^{4}+\frac{2}{5}a^{3}-\frac{9}{25}a^{2}-\frac{2}{5}a$, $\frac{1}{625}a^{23}-\frac{1}{625}a^{20}+\frac{2}{625}a^{19}-\frac{8}{625}a^{18}-\frac{7}{625}a^{17}-\frac{1}{625}a^{16}-\frac{9}{625}a^{15}+\frac{3}{625}a^{14}+\frac{4}{125}a^{13}-\frac{53}{625}a^{12}+\frac{26}{625}a^{11}+\frac{12}{625}a^{10}+\frac{51}{625}a^{9}+\frac{34}{625}a^{8}-\frac{91}{625}a^{7}+\frac{222}{625}a^{6}+\frac{8}{125}a^{5}+\frac{6}{25}a^{4}+\frac{11}{25}a^{3}+\frac{3}{25}a^{2}+\frac{1}{5}a$, $\frac{1}{6875}a^{24}-\frac{2}{6875}a^{23}+\frac{1}{6875}a^{22}+\frac{21}{6875}a^{21}-\frac{27}{6875}a^{20}-\frac{1}{1375}a^{19}-\frac{2}{125}a^{18}-\frac{63}{6875}a^{17}-\frac{1}{625}a^{16}+\frac{17}{6875}a^{15}-\frac{93}{6875}a^{14}+\frac{481}{6875}a^{13}-\frac{109}{1375}a^{12}+\frac{263}{6875}a^{11}+\frac{79}{6875}a^{10}+\frac{353}{6875}a^{9}-\frac{521}{6875}a^{8}+\frac{1224}{6875}a^{7}+\frac{569}{1375}a^{6}-\frac{394}{1375}a^{5}-\frac{11}{25}a^{4}+\frac{6}{25}a^{3}-\frac{9}{55}a^{2}+\frac{4}{55}a+\frac{2}{11}$, $\frac{1}{34375}a^{25}-\frac{1}{34375}a^{24}-\frac{1}{34375}a^{23}+\frac{2}{3125}a^{22}-\frac{61}{34375}a^{21}-\frac{87}{34375}a^{20}-\frac{23}{6875}a^{19}-\frac{448}{34375}a^{18}-\frac{349}{34375}a^{17}+\frac{501}{34375}a^{16}+\frac{144}{34375}a^{15}+\frac{388}{34375}a^{14}+\frac{1586}{34375}a^{13}+\frac{1368}{34375}a^{12}-\frac{2463}{34375}a^{11}+\frac{1477}{34375}a^{10}+\frac{3132}{34375}a^{9}+\frac{2903}{34375}a^{8}+\frac{3244}{34375}a^{7}+\frac{1814}{6875}a^{6}-\frac{411}{1375}a^{5}-\frac{18}{125}a^{4}+\frac{186}{1375}a^{3}-\frac{126}{275}a^{2}+\frac{27}{55}a-\frac{4}{11}$, $\frac{1}{26778125}a^{26}+\frac{1}{130625}a^{25}+\frac{543}{26778125}a^{24}-\frac{6084}{26778125}a^{23}+\frac{9016}{26778125}a^{22}-\frac{100688}{26778125}a^{21}+\frac{82493}{26778125}a^{20}+\frac{100977}{26778125}a^{19}+\frac{388008}{26778125}a^{18}+\frac{303262}{26778125}a^{17}-\frac{32526}{5355625}a^{16}-\frac{342528}{26778125}a^{15}-\frac{327476}{26778125}a^{14}-\frac{850011}{26778125}a^{13}-\frac{195039}{5355625}a^{12}+\frac{2129609}{26778125}a^{11}+\frac{1035609}{26778125}a^{10}-\frac{10898}{281875}a^{9}-\frac{374853}{26778125}a^{8}-\frac{13261541}{26778125}a^{7}+\frac{2183}{5225}a^{6}-\frac{452668}{1071125}a^{5}+\frac{388508}{1071125}a^{4}-\frac{35749}{97375}a^{3}+\frac{3148}{19475}a^{2}+\frac{11653}{42845}a-\frac{2570}{8569}$, $\frac{1}{68\!\cdots\!25}a^{27}+\frac{91\!\cdots\!62}{68\!\cdots\!25}a^{26}+\frac{73\!\cdots\!81}{68\!\cdots\!25}a^{25}+\frac{48\!\cdots\!69}{68\!\cdots\!25}a^{24}+\frac{92\!\cdots\!98}{13\!\cdots\!25}a^{23}-\frac{48\!\cdots\!69}{13\!\cdots\!25}a^{22}+\frac{13\!\cdots\!19}{68\!\cdots\!25}a^{21}-\frac{11\!\cdots\!03}{68\!\cdots\!25}a^{20}+\frac{21\!\cdots\!97}{68\!\cdots\!25}a^{19}-\frac{50\!\cdots\!91}{68\!\cdots\!25}a^{18}+\frac{11\!\cdots\!32}{68\!\cdots\!25}a^{17}-\frac{27\!\cdots\!71}{13\!\cdots\!25}a^{16}-\frac{21\!\cdots\!19}{13\!\cdots\!25}a^{15}+\frac{24\!\cdots\!61}{15\!\cdots\!75}a^{14}+\frac{29\!\cdots\!71}{61\!\cdots\!75}a^{13}-\frac{29\!\cdots\!47}{68\!\cdots\!25}a^{12}+\frac{44\!\cdots\!78}{68\!\cdots\!25}a^{11}-\frac{45\!\cdots\!26}{68\!\cdots\!25}a^{10}-\frac{59\!\cdots\!77}{68\!\cdots\!25}a^{9}-\frac{23\!\cdots\!28}{68\!\cdots\!25}a^{8}+\frac{83\!\cdots\!03}{13\!\cdots\!25}a^{7}+\frac{91\!\cdots\!78}{24\!\cdots\!75}a^{6}-\frac{25\!\cdots\!78}{27\!\cdots\!25}a^{5}+\frac{73\!\cdots\!33}{27\!\cdots\!25}a^{4}+\frac{89\!\cdots\!21}{54\!\cdots\!25}a^{3}-\frac{18\!\cdots\!36}{10\!\cdots\!25}a^{2}+\frac{33\!\cdots\!04}{21\!\cdots\!85}a-\frac{20\!\cdots\!71}{43\!\cdots\!97}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/5*a^8 + 2/5*a^7 - 2/5*a^6 - 1/5*a^5 - 2/5*a^4 + 2/5*a^3 + 1/5*a^2, 1/5*a^9 - 1/5*a^7 - 2/5*a^6 + 1/5*a^4 + 2/5*a^3 - 2/5*a^2, 1/5*a^10 - 2/5*a^6 + 1/5*a^2, 1/5*a^11 - 2/5*a^7 + 1/5*a^3, 1/5*a^12 - 1/5*a^7 + 1/5*a^6 - 2/5*a^5 + 2/5*a^4 - 1/5*a^3 + 2/5*a^2, 1/5*a^13 - 2/5*a^7 + 1/5*a^6 + 1/5*a^5 + 2/5*a^4 - 1/5*a^3 + 1/5*a^2, 1/25*a^14 + 2/25*a^12 + 2/25*a^11 + 2/25*a^10 + 1/25*a^9 - 1/25*a^8 + 1/25*a^7 + 1/5*a^6 - 3/25*a^5 - 3/25*a^4 - 1/5*a^3 - 2/5*a^2, 1/25*a^15 + 2/25*a^13 + 2/25*a^12 + 2/25*a^11 + 1/25*a^10 - 1/25*a^9 + 1/25*a^8 + 1/5*a^7 - 3/25*a^6 - 3/25*a^5 - 1/5*a^4 - 2/5*a^3, 1/25*a^16 + 2/25*a^13 - 2/25*a^12 + 2/25*a^11 - 1/25*a^9 + 2/25*a^8 + 12/25*a^6 + 6/25*a^5 + 6/25*a^4 + 1/5*a^3 - 1/5*a^2, 1/25*a^17 - 2/25*a^13 - 2/25*a^12 + 1/25*a^11 + 2/25*a^8 + 11/25*a^6 + 12/25*a^5 + 11/25*a^4 + 2/5*a^3, 1/25*a^18 - 2/25*a^13 - 1/25*a^11 - 1/25*a^10 - 1/25*a^9 - 2/25*a^8 + 8/25*a^7 + 12/25*a^6 - 2/5*a^5 - 11/25*a^4 + 1/5*a^3, 1/125*a^19 - 2/125*a^18 - 1/125*a^17 + 2/125*a^16 + 1/125*a^15 - 1/125*a^14 + 7/125*a^13 + 6/125*a^12 + 8/125*a^11 + 9/125*a^10 + 3/125*a^9 + 9/125*a^8 - 8/125*a^7 - 29/125*a^6 - 7/125*a^5 - 3/25*a^4 - 1/5*a^3 + 2/5*a^2 + 2/5*a, 1/125*a^20 + 1/125*a^15 - 1/25*a^12 + 6/125*a^10 + 2/25*a^9 - 1/25*a^8 + 3/25*a^7 + 2/25*a^6 - 44/125*a^5 - 2/5*a^4 + 2/5*a^3 + 2/5*a^2 - 1/5*a, 1/125*a^21 + 1/125*a^16 - 1/25*a^13 + 6/125*a^11 + 2/25*a^10 - 1/25*a^9 - 2/25*a^8 - 8/25*a^7 + 6/125*a^6 - 1/5*a^5 - 1/5*a^4 - 2/5*a^2, 1/625*a^22 + 2/625*a^21 - 1/625*a^20 + 2/625*a^19 - 9/625*a^18 + 4/625*a^17 - 9/625*a^16 - 4/625*a^15 - 2/625*a^14 + 39/625*a^13 - 32/625*a^12 - 57/625*a^11 + 62/625*a^10 + 56/625*a^9 + 18/625*a^8 + 2/125*a^7 + 99/625*a^6 + 1/5*a^5 + 11/25*a^4 + 2/5*a^3 - 9/25*a^2 - 2/5*a, 1/625*a^23 - 1/625*a^20 + 2/625*a^19 - 8/625*a^18 - 7/625*a^17 - 1/625*a^16 - 9/625*a^15 + 3/625*a^14 + 4/125*a^13 - 53/625*a^12 + 26/625*a^11 + 12/625*a^10 + 51/625*a^9 + 34/625*a^8 - 91/625*a^7 + 222/625*a^6 + 8/125*a^5 + 6/25*a^4 + 11/25*a^3 + 3/25*a^2 + 1/5*a, 1/6875*a^24 - 2/6875*a^23 + 1/6875*a^22 + 21/6875*a^21 - 27/6875*a^20 - 1/1375*a^19 - 2/125*a^18 - 63/6875*a^17 - 1/625*a^16 + 17/6875*a^15 - 93/6875*a^14 + 481/6875*a^13 - 109/1375*a^12 + 263/6875*a^11 + 79/6875*a^10 + 353/6875*a^9 - 521/6875*a^8 + 1224/6875*a^7 + 569/1375*a^6 - 394/1375*a^5 - 11/25*a^4 + 6/25*a^3 - 9/55*a^2 + 4/55*a + 2/11, 1/34375*a^25 - 1/34375*a^24 - 1/34375*a^23 + 2/3125*a^22 - 61/34375*a^21 - 87/34375*a^20 - 23/6875*a^19 - 448/34375*a^18 - 349/34375*a^17 + 501/34375*a^16 + 144/34375*a^15 + 388/34375*a^14 + 1586/34375*a^13 + 1368/34375*a^12 - 2463/34375*a^11 + 1477/34375*a^10 + 3132/34375*a^9 + 2903/34375*a^8 + 3244/34375*a^7 + 1814/6875*a^6 - 411/1375*a^5 - 18/125*a^4 + 186/1375*a^3 - 126/275*a^2 + 27/55*a - 4/11, 1/26778125*a^26 + 1/130625*a^25 + 543/26778125*a^24 - 6084/26778125*a^23 + 9016/26778125*a^22 - 100688/26778125*a^21 + 82493/26778125*a^20 + 100977/26778125*a^19 + 388008/26778125*a^18 + 303262/26778125*a^17 - 32526/5355625*a^16 - 342528/26778125*a^15 - 327476/26778125*a^14 - 850011/26778125*a^13 - 195039/5355625*a^12 + 2129609/26778125*a^11 + 1035609/26778125*a^10 - 10898/281875*a^9 - 374853/26778125*a^8 - 13261541/26778125*a^7 + 2183/5225*a^6 - 452668/1071125*a^5 + 388508/1071125*a^4 - 35749/97375*a^3 + 3148/19475*a^2 + 11653/42845*a - 2570/8569, 1/68197585161567464382778623695847273894144760890625*a^27 + 919207749021906554682959668295283150569862/68197585161567464382778623695847273894144760890625*a^26 + 734587698440445715999305411999321280897312481/68197585161567464382778623695847273894144760890625*a^25 + 4826882319811856074156454294079503111987396969/68197585161567464382778623695847273894144760890625*a^24 + 9207646722489444534181629106955156618791364198/13639517032313492876555724739169454778828952178125*a^23 - 4875381967358295922385459078525796834712883369/13639517032313492876555724739169454778828952178125*a^22 + 132187126730934484308749726582853353476167653519/68197585161567464382778623695847273894144760890625*a^21 - 11324727442892648458778821260631200711883032103/68197585161567464382778623695847273894144760890625*a^20 + 213929358529516500173946625642667993254645629097/68197585161567464382778623695847273894144760890625*a^19 - 506909480747572256568579375713705106032137010291/68197585161567464382778623695847273894144760890625*a^18 + 1194213751807359788305778752747583407336431995932/68197585161567464382778623695847273894144760890625*a^17 - 271632935331935729201633489422462222123468811671/13639517032313492876555724739169454778828952178125*a^16 - 217894135214727561193887412043839427492668917019/13639517032313492876555724739169454778828952178125*a^15 + 242909529162515945579326889464479996912842161/151214157786180630560484753205869786904977296875*a^14 + 297774521533828449884671967739323912927228476971/6199780469233405852979874881440661263104069171875*a^13 - 2920889944079587267285758245807321029207345104447/68197585161567464382778623695847273894144760890625*a^12 + 4435536312246787018071535765022570441261712245678/68197585161567464382778623695847273894144760890625*a^11 - 4585611167609831281935250915217012903022670604826/68197585161567464382778623695847273894144760890625*a^10 - 5929001284063086406005545392381870398294116416377/68197585161567464382778623695847273894144760890625*a^9 - 2308755217309046394643773359472379715905965540628/68197585161567464382778623695847273894144760890625*a^8 + 837020560450204070127701393019846024785301543903/13639517032313492876555724739169454778828952178125*a^7 + 91757338010360042892384289280444312577421781178/247991218769336234119194995257626450524162766875*a^6 - 25163566944880033015224760979024859569205484878/2727903406462698575311144947833890955765790435625*a^5 + 731243363267716860050651594519464739000747358633/2727903406462698575311144947833890955765790435625*a^4 + 89361547003539180550559924511010001672313391821/545580681292539715062228989566778191153158087125*a^3 - 18714866590684908613001126609036829179188699936/109116136258507943012445797913355638230631617425*a^2 + 3313184066430435527920984601531430103617420404/21823227251701588602489159582671127646126323485*a - 2094308493428239412423457222950719576375792571/4364645450340317720497831916534225529225264697

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$5$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$14$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{53\!\cdots\!49}{27\!\cdots\!75}a^{27}-\frac{19\!\cdots\!97}{27\!\cdots\!75}a^{26}-\frac{46\!\cdots\!51}{27\!\cdots\!75}a^{25}+\frac{25\!\cdots\!96}{27\!\cdots\!75}a^{24}+\frac{58\!\cdots\!89}{54\!\cdots\!75}a^{23}-\frac{40\!\cdots\!86}{54\!\cdots\!75}a^{22}-\frac{12\!\cdots\!69}{27\!\cdots\!75}a^{21}+\frac{10\!\cdots\!38}{27\!\cdots\!75}a^{20}+\frac{68\!\cdots\!33}{27\!\cdots\!75}a^{19}-\frac{37\!\cdots\!29}{27\!\cdots\!75}a^{18}-\frac{13\!\cdots\!22}{27\!\cdots\!75}a^{17}+\frac{19\!\cdots\!82}{54\!\cdots\!75}a^{16}-\frac{18\!\cdots\!76}{54\!\cdots\!75}a^{15}-\frac{23\!\cdots\!36}{27\!\cdots\!75}a^{14}+\frac{39\!\cdots\!59}{27\!\cdots\!75}a^{13}-\frac{84\!\cdots\!13}{27\!\cdots\!75}a^{12}+\frac{36\!\cdots\!42}{27\!\cdots\!75}a^{11}+\frac{32\!\cdots\!71}{27\!\cdots\!75}a^{10}-\frac{78\!\cdots\!88}{27\!\cdots\!75}a^{9}+\frac{19\!\cdots\!23}{27\!\cdots\!75}a^{8}+\frac{54\!\cdots\!13}{54\!\cdots\!75}a^{7}-\frac{20\!\cdots\!92}{10\!\cdots\!75}a^{6}+\frac{22\!\cdots\!93}{10\!\cdots\!75}a^{5}+\frac{95\!\cdots\!72}{10\!\cdots\!75}a^{4}-\frac{43\!\cdots\!72}{21\!\cdots\!75}a^{3}+\frac{37\!\cdots\!77}{43\!\cdots\!75}a^{2}-\frac{51\!\cdots\!22}{87\!\cdots\!15}a-\frac{23\!\cdots\!74}{17\!\cdots\!83}$, $\frac{37\!\cdots\!96}{68\!\cdots\!25}a^{27}-\frac{16\!\cdots\!18}{68\!\cdots\!25}a^{26}-\frac{20\!\cdots\!09}{61\!\cdots\!75}a^{25}+\frac{20\!\cdots\!14}{68\!\cdots\!25}a^{24}+\frac{17\!\cdots\!64}{13\!\cdots\!25}a^{23}-\frac{31\!\cdots\!73}{13\!\cdots\!25}a^{22}-\frac{46\!\cdots\!91}{68\!\cdots\!25}a^{21}+\frac{82\!\cdots\!02}{68\!\cdots\!25}a^{20}+\frac{96\!\cdots\!72}{68\!\cdots\!25}a^{19}-\frac{31\!\cdots\!71}{68\!\cdots\!25}a^{18}+\frac{47\!\cdots\!78}{35\!\cdots\!75}a^{17}+\frac{14\!\cdots\!24}{13\!\cdots\!25}a^{16}-\frac{16\!\cdots\!77}{13\!\cdots\!25}a^{15}-\frac{13\!\cdots\!74}{68\!\cdots\!25}a^{14}+\frac{33\!\cdots\!46}{68\!\cdots\!25}a^{13}-\frac{14\!\cdots\!12}{68\!\cdots\!25}a^{12}+\frac{46\!\cdots\!63}{16\!\cdots\!25}a^{11}-\frac{17\!\cdots\!51}{68\!\cdots\!25}a^{10}-\frac{55\!\cdots\!92}{68\!\cdots\!25}a^{9}+\frac{24\!\cdots\!97}{68\!\cdots\!25}a^{8}+\frac{38\!\cdots\!78}{71\!\cdots\!75}a^{7}+\frac{34\!\cdots\!81}{27\!\cdots\!25}a^{6}+\frac{92\!\cdots\!82}{27\!\cdots\!25}a^{5}-\frac{89\!\cdots\!67}{27\!\cdots\!25}a^{4}-\frac{10\!\cdots\!88}{13\!\cdots\!25}a^{3}+\frac{36\!\cdots\!04}{10\!\cdots\!25}a^{2}+\frac{17\!\cdots\!71}{21\!\cdots\!85}a+\frac{14\!\cdots\!21}{43\!\cdots\!97}$, $\frac{31\!\cdots\!14}{10\!\cdots\!75}a^{27}-\frac{68\!\cdots\!33}{53\!\cdots\!25}a^{26}-\frac{20\!\cdots\!21}{10\!\cdots\!75}a^{25}+\frac{16\!\cdots\!61}{10\!\cdots\!75}a^{24}+\frac{20\!\cdots\!14}{20\!\cdots\!75}a^{23}-\frac{25\!\cdots\!17}{20\!\cdots\!75}a^{22}-\frac{28\!\cdots\!19}{10\!\cdots\!75}a^{21}+\frac{64\!\cdots\!58}{10\!\cdots\!75}a^{20}+\frac{21\!\cdots\!93}{10\!\cdots\!75}a^{19}-\frac{23\!\cdots\!44}{10\!\cdots\!75}a^{18}+\frac{30\!\cdots\!28}{10\!\cdots\!75}a^{17}+\frac{21\!\cdots\!32}{40\!\cdots\!75}a^{16}-\frac{11\!\cdots\!53}{20\!\cdots\!75}a^{15}-\frac{10\!\cdots\!41}{10\!\cdots\!75}a^{14}+\frac{24\!\cdots\!49}{10\!\cdots\!75}a^{13}-\frac{10\!\cdots\!98}{10\!\cdots\!75}a^{12}+\frac{52\!\cdots\!47}{24\!\cdots\!75}a^{11}-\frac{93\!\cdots\!79}{10\!\cdots\!75}a^{10}-\frac{47\!\cdots\!88}{10\!\cdots\!75}a^{9}+\frac{10\!\cdots\!48}{92\!\cdots\!25}a^{8}+\frac{10\!\cdots\!62}{20\!\cdots\!75}a^{7}+\frac{71\!\cdots\!14}{81\!\cdots\!75}a^{6}+\frac{13\!\cdots\!28}{40\!\cdots\!75}a^{5}-\frac{28\!\cdots\!08}{40\!\cdots\!75}a^{4}-\frac{45\!\cdots\!21}{19\!\cdots\!75}a^{3}+\frac{14\!\cdots\!33}{16\!\cdots\!75}a^{2}-\frac{37\!\cdots\!85}{65\!\cdots\!91}a+\frac{33\!\cdots\!41}{65\!\cdots\!91}$, $\frac{70\!\cdots\!84}{68\!\cdots\!25}a^{27}-\frac{29\!\cdots\!12}{68\!\cdots\!25}a^{26}-\frac{47\!\cdots\!31}{68\!\cdots\!25}a^{25}+\frac{36\!\cdots\!46}{68\!\cdots\!25}a^{24}+\frac{89\!\cdots\!06}{27\!\cdots\!25}a^{23}-\frac{13\!\cdots\!49}{33\!\cdots\!25}a^{22}-\frac{56\!\cdots\!84}{68\!\cdots\!25}a^{21}+\frac{15\!\cdots\!83}{68\!\cdots\!25}a^{20}+\frac{41\!\cdots\!83}{68\!\cdots\!25}a^{19}-\frac{56\!\cdots\!74}{68\!\cdots\!25}a^{18}+\frac{94\!\cdots\!13}{68\!\cdots\!25}a^{17}+\frac{26\!\cdots\!19}{13\!\cdots\!25}a^{16}-\frac{28\!\cdots\!02}{13\!\cdots\!25}a^{15}-\frac{24\!\cdots\!26}{61\!\cdots\!75}a^{14}+\frac{58\!\cdots\!64}{68\!\cdots\!25}a^{13}-\frac{18\!\cdots\!03}{68\!\cdots\!25}a^{12}+\frac{30\!\cdots\!02}{68\!\cdots\!25}a^{11}-\frac{21\!\cdots\!09}{68\!\cdots\!25}a^{10}-\frac{98\!\cdots\!38}{68\!\cdots\!25}a^{9}+\frac{41\!\cdots\!03}{68\!\cdots\!25}a^{8}+\frac{55\!\cdots\!28}{13\!\cdots\!25}a^{7}+\frac{94\!\cdots\!68}{27\!\cdots\!25}a^{6}+\frac{22\!\cdots\!43}{27\!\cdots\!25}a^{5}-\frac{13\!\cdots\!83}{27\!\cdots\!25}a^{4}-\frac{51\!\cdots\!52}{54\!\cdots\!25}a^{3}+\frac{35\!\cdots\!67}{10\!\cdots\!25}a^{2}-\frac{31\!\cdots\!02}{21\!\cdots\!85}a+\frac{85\!\cdots\!53}{43\!\cdots\!97}$, $\frac{25\!\cdots\!16}{13\!\cdots\!25}a^{27}-\frac{89\!\cdots\!26}{54\!\cdots\!25}a^{26}+\frac{10\!\cdots\!74}{27\!\cdots\!25}a^{25}+\frac{13\!\cdots\!34}{13\!\cdots\!25}a^{24}-\frac{64\!\cdots\!61}{13\!\cdots\!25}a^{23}-\frac{48\!\cdots\!19}{13\!\cdots\!25}a^{22}+\frac{12\!\cdots\!91}{33\!\cdots\!25}a^{21}-\frac{58\!\cdots\!27}{13\!\cdots\!25}a^{20}-\frac{24\!\cdots\!92}{13\!\cdots\!25}a^{19}+\frac{80\!\cdots\!46}{13\!\cdots\!25}a^{18}+\frac{94\!\cdots\!17}{12\!\cdots\!75}a^{17}-\frac{96\!\cdots\!71}{13\!\cdots\!25}a^{16}-\frac{23\!\cdots\!33}{13\!\cdots\!25}a^{15}+\frac{16\!\cdots\!99}{54\!\cdots\!25}a^{14}+\frac{27\!\cdots\!22}{13\!\cdots\!25}a^{13}-\frac{31\!\cdots\!79}{27\!\cdots\!25}a^{12}+\frac{18\!\cdots\!23}{13\!\cdots\!25}a^{11}-\frac{15\!\cdots\!21}{13\!\cdots\!25}a^{10}+\frac{14\!\cdots\!11}{13\!\cdots\!25}a^{9}+\frac{42\!\cdots\!41}{60\!\cdots\!75}a^{8}-\frac{27\!\cdots\!57}{13\!\cdots\!25}a^{7}+\frac{16\!\cdots\!23}{27\!\cdots\!25}a^{6}-\frac{53\!\cdots\!47}{10\!\cdots\!25}a^{5}+\frac{16\!\cdots\!81}{10\!\cdots\!25}a^{4}+\frac{44\!\cdots\!82}{54\!\cdots\!25}a^{3}+\frac{54\!\cdots\!93}{10\!\cdots\!25}a^{2}-\frac{33\!\cdots\!81}{39\!\cdots\!27}a-\frac{30\!\cdots\!84}{43\!\cdots\!97}$, $\frac{99\!\cdots\!52}{68\!\cdots\!25}a^{27}+\frac{22\!\cdots\!04}{16\!\cdots\!25}a^{26}-\frac{27\!\cdots\!23}{68\!\cdots\!25}a^{25}-\frac{95\!\cdots\!82}{68\!\cdots\!25}a^{24}+\frac{72\!\cdots\!81}{13\!\cdots\!25}a^{23}+\frac{15\!\cdots\!01}{13\!\cdots\!25}a^{22}-\frac{27\!\cdots\!72}{68\!\cdots\!25}a^{21}-\frac{43\!\cdots\!91}{68\!\cdots\!25}a^{20}+\frac{72\!\cdots\!96}{35\!\cdots\!75}a^{19}+\frac{23\!\cdots\!43}{68\!\cdots\!25}a^{18}-\frac{36\!\cdots\!56}{61\!\cdots\!75}a^{17}-\frac{78\!\cdots\!84}{13\!\cdots\!25}a^{16}+\frac{45\!\cdots\!37}{27\!\cdots\!25}a^{15}-\frac{39\!\cdots\!48}{68\!\cdots\!25}a^{14}-\frac{38\!\cdots\!88}{68\!\cdots\!25}a^{13}+\frac{12\!\cdots\!29}{35\!\cdots\!75}a^{12}+\frac{93\!\cdots\!71}{68\!\cdots\!25}a^{11}+\frac{48\!\cdots\!88}{68\!\cdots\!25}a^{10}+\frac{49\!\cdots\!01}{68\!\cdots\!25}a^{9}-\frac{73\!\cdots\!81}{68\!\cdots\!25}a^{8}-\frac{39\!\cdots\!28}{66\!\cdots\!25}a^{7}-\frac{13\!\cdots\!56}{49\!\cdots\!75}a^{6}-\frac{11\!\cdots\!61}{27\!\cdots\!25}a^{5}+\frac{21\!\cdots\!41}{27\!\cdots\!25}a^{4}+\frac{31\!\cdots\!63}{54\!\cdots\!25}a^{3}-\frac{47\!\cdots\!83}{10\!\cdots\!25}a^{2}+\frac{40\!\cdots\!96}{21\!\cdots\!85}a-\frac{55\!\cdots\!56}{10\!\cdots\!17}$, $\frac{42\!\cdots\!29}{13\!\cdots\!25}a^{27}-\frac{18\!\cdots\!81}{13\!\cdots\!25}a^{26}-\frac{24\!\cdots\!32}{13\!\cdots\!25}a^{25}+\frac{45\!\cdots\!71}{27\!\cdots\!25}a^{24}+\frac{91\!\cdots\!12}{13\!\cdots\!25}a^{23}-\frac{17\!\cdots\!26}{13\!\cdots\!25}a^{22}-\frac{23\!\cdots\!21}{13\!\cdots\!25}a^{21}+\frac{81\!\cdots\!08}{12\!\cdots\!75}a^{20}+\frac{18\!\cdots\!81}{27\!\cdots\!25}a^{19}-\frac{33\!\cdots\!53}{13\!\cdots\!25}a^{18}+\frac{11\!\cdots\!59}{13\!\cdots\!25}a^{17}+\frac{75\!\cdots\!84}{13\!\cdots\!25}a^{16}-\frac{96\!\cdots\!92}{13\!\cdots\!25}a^{15}-\frac{99\!\cdots\!13}{99\!\cdots\!75}a^{14}+\frac{36\!\cdots\!17}{13\!\cdots\!25}a^{13}-\frac{19\!\cdots\!01}{13\!\cdots\!25}a^{12}+\frac{26\!\cdots\!34}{13\!\cdots\!25}a^{11}-\frac{16\!\cdots\!47}{12\!\cdots\!75}a^{10}-\frac{29\!\cdots\!44}{66\!\cdots\!25}a^{9}+\frac{35\!\cdots\!02}{13\!\cdots\!25}a^{8}+\frac{27\!\cdots\!13}{27\!\cdots\!25}a^{7}+\frac{20\!\cdots\!31}{27\!\cdots\!25}a^{6}+\frac{15\!\cdots\!19}{54\!\cdots\!25}a^{5}-\frac{10\!\cdots\!32}{54\!\cdots\!25}a^{4}-\frac{32\!\cdots\!54}{10\!\cdots\!25}a^{3}+\frac{13\!\cdots\!59}{10\!\cdots\!25}a^{2}-\frac{89\!\cdots\!97}{21\!\cdots\!85}a+\frac{31\!\cdots\!61}{43\!\cdots\!97}$, $\frac{71\!\cdots\!38}{68\!\cdots\!25}a^{27}-\frac{30\!\cdots\!39}{68\!\cdots\!25}a^{26}-\frac{45\!\cdots\!32}{68\!\cdots\!25}a^{25}+\frac{37\!\cdots\!97}{68\!\cdots\!25}a^{24}+\frac{40\!\cdots\!32}{13\!\cdots\!25}a^{23}-\frac{11\!\cdots\!47}{27\!\cdots\!25}a^{22}-\frac{40\!\cdots\!33}{68\!\cdots\!25}a^{21}+\frac{14\!\cdots\!96}{68\!\cdots\!25}a^{20}+\frac{34\!\cdots\!46}{68\!\cdots\!25}a^{19}-\frac{49\!\cdots\!58}{61\!\cdots\!75}a^{18}+\frac{11\!\cdots\!81}{61\!\cdots\!75}a^{17}+\frac{49\!\cdots\!38}{27\!\cdots\!25}a^{16}-\frac{26\!\cdots\!08}{12\!\cdots\!75}a^{15}-\frac{24\!\cdots\!42}{68\!\cdots\!25}a^{14}+\frac{57\!\cdots\!88}{68\!\cdots\!25}a^{13}-\frac{28\!\cdots\!16}{68\!\cdots\!25}a^{12}+\frac{45\!\cdots\!14}{68\!\cdots\!25}a^{11}-\frac{25\!\cdots\!88}{68\!\cdots\!25}a^{10}-\frac{91\!\cdots\!86}{61\!\cdots\!75}a^{9}+\frac{47\!\cdots\!16}{68\!\cdots\!25}a^{8}+\frac{80\!\cdots\!48}{27\!\cdots\!25}a^{7}+\frac{72\!\cdots\!69}{27\!\cdots\!25}a^{6}+\frac{25\!\cdots\!56}{27\!\cdots\!25}a^{5}-\frac{13\!\cdots\!66}{24\!\cdots\!75}a^{4}-\frac{51\!\cdots\!43}{54\!\cdots\!25}a^{3}+\frac{36\!\cdots\!49}{10\!\cdots\!25}a^{2}-\frac{31\!\cdots\!38}{21\!\cdots\!85}a+\frac{98\!\cdots\!89}{43\!\cdots\!97}$, $\frac{66\!\cdots\!56}{13\!\cdots\!25}a^{27}-\frac{12\!\cdots\!88}{71\!\cdots\!75}a^{26}-\frac{58\!\cdots\!97}{13\!\cdots\!25}a^{25}+\frac{66\!\cdots\!41}{27\!\cdots\!25}a^{24}+\frac{35\!\cdots\!54}{13\!\cdots\!25}a^{23}-\frac{52\!\cdots\!01}{27\!\cdots\!25}a^{22}-\frac{15\!\cdots\!56}{13\!\cdots\!25}a^{21}+\frac{13\!\cdots\!51}{13\!\cdots\!25}a^{20}+\frac{88\!\cdots\!49}{13\!\cdots\!25}a^{19}-\frac{49\!\cdots\!99}{13\!\cdots\!25}a^{18}-\frac{70\!\cdots\!59}{13\!\cdots\!25}a^{17}+\frac{12\!\cdots\!72}{13\!\cdots\!25}a^{16}-\frac{21\!\cdots\!88}{27\!\cdots\!25}a^{15}-\frac{24\!\cdots\!39}{12\!\cdots\!75}a^{14}+\frac{48\!\cdots\!12}{13\!\cdots\!25}a^{13}-\frac{98\!\cdots\!16}{13\!\cdots\!25}a^{12}+\frac{16\!\cdots\!16}{12\!\cdots\!75}a^{11}+\frac{52\!\cdots\!37}{13\!\cdots\!25}a^{10}-\frac{18\!\cdots\!73}{13\!\cdots\!25}a^{9}+\frac{21\!\cdots\!98}{27\!\cdots\!25}a^{8}-\frac{62\!\cdots\!43}{13\!\cdots\!25}a^{7}+\frac{49\!\cdots\!59}{27\!\cdots\!25}a^{6}+\frac{21\!\cdots\!91}{21\!\cdots\!85}a^{5}-\frac{74\!\cdots\!02}{10\!\cdots\!25}a^{4}+\frac{19\!\cdots\!68}{54\!\cdots\!25}a^{3}-\frac{92\!\cdots\!88}{21\!\cdots\!85}a^{2}+\frac{48\!\cdots\!36}{21\!\cdots\!85}a-\frac{28\!\cdots\!72}{43\!\cdots\!97}$, $\frac{60\!\cdots\!02}{16\!\cdots\!25}a^{27}-\frac{10\!\cdots\!86}{68\!\cdots\!25}a^{26}-\frac{41\!\cdots\!68}{16\!\cdots\!25}a^{25}+\frac{12\!\cdots\!53}{68\!\cdots\!25}a^{24}+\frac{17\!\cdots\!98}{13\!\cdots\!25}a^{23}-\frac{19\!\cdots\!79}{13\!\cdots\!25}a^{22}-\frac{27\!\cdots\!02}{68\!\cdots\!25}a^{21}+\frac{49\!\cdots\!29}{68\!\cdots\!25}a^{20}+\frac{98\!\cdots\!56}{35\!\cdots\!75}a^{19}-\frac{16\!\cdots\!12}{61\!\cdots\!75}a^{18}+\frac{21\!\cdots\!29}{68\!\cdots\!25}a^{17}+\frac{76\!\cdots\!52}{12\!\cdots\!75}a^{16}-\frac{18\!\cdots\!97}{27\!\cdots\!25}a^{15}-\frac{87\!\cdots\!68}{68\!\cdots\!25}a^{14}+\frac{18\!\cdots\!82}{68\!\cdots\!25}a^{13}-\frac{41\!\cdots\!01}{35\!\cdots\!75}a^{12}+\frac{15\!\cdots\!31}{68\!\cdots\!25}a^{11}-\frac{58\!\cdots\!72}{68\!\cdots\!25}a^{10}-\frac{34\!\cdots\!49}{68\!\cdots\!25}a^{9}+\frac{11\!\cdots\!49}{68\!\cdots\!25}a^{8}+\frac{11\!\cdots\!16}{13\!\cdots\!25}a^{7}+\frac{44\!\cdots\!48}{60\!\cdots\!75}a^{6}+\frac{95\!\cdots\!44}{27\!\cdots\!25}a^{5}-\frac{31\!\cdots\!14}{27\!\cdots\!25}a^{4}-\frac{16\!\cdots\!83}{54\!\cdots\!25}a^{3}+\frac{94\!\cdots\!51}{10\!\cdots\!25}a^{2}-\frac{13\!\cdots\!79}{21\!\cdots\!85}a+\frac{24\!\cdots\!06}{43\!\cdots\!97}$, $\frac{59\!\cdots\!36}{68\!\cdots\!25}a^{27}-\frac{13\!\cdots\!68}{68\!\cdots\!25}a^{26}-\frac{86\!\cdots\!89}{68\!\cdots\!25}a^{25}+\frac{22\!\cdots\!89}{68\!\cdots\!25}a^{24}+\frac{15\!\cdots\!24}{13\!\cdots\!25}a^{23}-\frac{34\!\cdots\!26}{12\!\cdots\!75}a^{22}-\frac{51\!\cdots\!61}{68\!\cdots\!25}a^{21}+\frac{10\!\cdots\!02}{68\!\cdots\!25}a^{20}+\frac{28\!\cdots\!42}{68\!\cdots\!25}a^{19}-\frac{35\!\cdots\!11}{68\!\cdots\!25}a^{18}-\frac{82\!\cdots\!53}{68\!\cdots\!25}a^{17}+\frac{22\!\cdots\!43}{13\!\cdots\!25}a^{16}+\frac{21\!\cdots\!92}{13\!\cdots\!25}a^{15}-\frac{43\!\cdots\!94}{68\!\cdots\!25}a^{14}-\frac{20\!\cdots\!14}{68\!\cdots\!25}a^{13}+\frac{67\!\cdots\!38}{61\!\cdots\!75}a^{12}+\frac{15\!\cdots\!23}{68\!\cdots\!25}a^{11}+\frac{16\!\cdots\!29}{68\!\cdots\!25}a^{10}-\frac{10\!\cdots\!32}{68\!\cdots\!25}a^{9}-\frac{10\!\cdots\!73}{68\!\cdots\!25}a^{8}+\frac{14\!\cdots\!84}{13\!\cdots\!25}a^{7}+\frac{19\!\cdots\!41}{27\!\cdots\!25}a^{6}+\frac{54\!\cdots\!72}{27\!\cdots\!25}a^{5}+\frac{26\!\cdots\!78}{27\!\cdots\!25}a^{4}-\frac{29\!\cdots\!68}{21\!\cdots\!85}a^{3}-\frac{82\!\cdots\!12}{10\!\cdots\!25}a^{2}-\frac{31\!\cdots\!99}{21\!\cdots\!85}a-\frac{10\!\cdots\!54}{43\!\cdots\!97}$, $\frac{15\!\cdots\!89}{68\!\cdots\!25}a^{27}-\frac{74\!\cdots\!72}{61\!\cdots\!75}a^{26}-\frac{37\!\cdots\!11}{68\!\cdots\!25}a^{25}+\frac{93\!\cdots\!56}{68\!\cdots\!25}a^{24}-\frac{74\!\cdots\!66}{13\!\cdots\!25}a^{23}-\frac{13\!\cdots\!96}{13\!\cdots\!25}a^{22}+\frac{55\!\cdots\!16}{68\!\cdots\!25}a^{21}+\frac{31\!\cdots\!13}{61\!\cdots\!75}a^{20}-\frac{26\!\cdots\!87}{68\!\cdots\!25}a^{19}-\frac{13\!\cdots\!69}{68\!\cdots\!25}a^{18}+\frac{14\!\cdots\!03}{61\!\cdots\!75}a^{17}+\frac{53\!\cdots\!02}{13\!\cdots\!25}a^{16}-\frac{12\!\cdots\!76}{13\!\cdots\!25}a^{15}-\frac{22\!\cdots\!96}{68\!\cdots\!25}a^{14}+\frac{19\!\cdots\!49}{68\!\cdots\!25}a^{13}-\frac{19\!\cdots\!18}{68\!\cdots\!25}a^{12}+\frac{12\!\cdots\!37}{68\!\cdots\!25}a^{11}-\frac{82\!\cdots\!94}{68\!\cdots\!25}a^{10}-\frac{24\!\cdots\!93}{68\!\cdots\!25}a^{9}+\frac{37\!\cdots\!28}{68\!\cdots\!25}a^{8}-\frac{13\!\cdots\!82}{13\!\cdots\!25}a^{7}-\frac{31\!\cdots\!86}{27\!\cdots\!25}a^{6}+\frac{73\!\cdots\!43}{27\!\cdots\!25}a^{5}-\frac{83\!\cdots\!08}{27\!\cdots\!25}a^{4}-\frac{69\!\cdots\!67}{54\!\cdots\!25}a^{3}+\frac{37\!\cdots\!56}{10\!\cdots\!25}a^{2}-\frac{33\!\cdots\!04}{21\!\cdots\!85}a+\frac{81\!\cdots\!37}{43\!\cdots\!97}$, $\frac{80\!\cdots\!06}{68\!\cdots\!25}a^{27}-\frac{36\!\cdots\!73}{68\!\cdots\!25}a^{26}-\frac{34\!\cdots\!29}{68\!\cdots\!25}a^{25}+\frac{40\!\cdots\!44}{68\!\cdots\!25}a^{24}+\frac{13\!\cdots\!77}{13\!\cdots\!25}a^{23}-\frac{58\!\cdots\!49}{13\!\cdots\!25}a^{22}+\frac{58\!\cdots\!89}{68\!\cdots\!25}a^{21}+\frac{12\!\cdots\!57}{61\!\cdots\!75}a^{20}-\frac{58\!\cdots\!33}{68\!\cdots\!25}a^{19}-\frac{48\!\cdots\!86}{68\!\cdots\!25}a^{18}+\frac{30\!\cdots\!62}{68\!\cdots\!25}a^{17}+\frac{88\!\cdots\!19}{71\!\cdots\!75}a^{16}-\frac{35\!\cdots\!14}{13\!\cdots\!25}a^{15}-\frac{10\!\cdots\!34}{68\!\cdots\!25}a^{14}+\frac{54\!\cdots\!16}{68\!\cdots\!25}a^{13}-\frac{69\!\cdots\!02}{68\!\cdots\!25}a^{12}+\frac{12\!\cdots\!83}{68\!\cdots\!25}a^{11}-\frac{84\!\cdots\!66}{68\!\cdots\!25}a^{10}-\frac{82\!\cdots\!42}{68\!\cdots\!25}a^{9}+\frac{48\!\cdots\!63}{35\!\cdots\!75}a^{8}-\frac{20\!\cdots\!33}{10\!\cdots\!25}a^{7}+\frac{31\!\cdots\!26}{14\!\cdots\!75}a^{6}+\frac{17\!\cdots\!07}{27\!\cdots\!25}a^{5}-\frac{25\!\cdots\!42}{27\!\cdots\!25}a^{4}+\frac{20\!\cdots\!99}{54\!\cdots\!25}a^{3}-\frac{88\!\cdots\!38}{10\!\cdots\!25}a^{2}-\frac{54\!\cdots\!98}{21\!\cdots\!85}a+\frac{22\!\cdots\!77}{43\!\cdots\!97}$, 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Regulator:	$ 2346058429844.763 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{13}\cdot 2346058429844.763 \cdot 1}{2\cdot\sqrt{725917312055025700242151478457681268310546875}}\cr\approx \mathstrut & 4.14250967749672 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$D_{28}$ (as 28T10):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 56

The 17 conjugacy class representatives for $D_{28}$

Character table for $D_{28}$

Intermediate fields

$\Q(\sqrt{5}) $, 4.2.12475.1, 7.1.15531437375.1, 14.2.1206127734667734453125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 28 sibling:	deg 28
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$28$	$28$	R	${\href{/padicField/7.4.0.1}{4} }^{7}$	${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$	$28$	$28$	${\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	$28$	${\href{/padicField/29.7.0.1}{7} }^{4}$	${\href{/padicField/31.7.0.1}{7} }^{4}$	$28$	${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.2.0.1}{2} }^{14}$	${\href{/padicField/47.2.0.1}{2} }^{14}$	$28$	${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$5$ 5	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$499$ 499	$\Q_{499}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{499}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.499.2t1.a.a	$1$	$ 499 $	$\Q(\sqrt{-499}) $	$C_2$ (as 2T1)	$1$	$-1$
	1.2495.2t1.a.a	$1$	$ 5 \cdot 499 $	$\Q(\sqrt{-2495}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.2t1.a.a	$1$	$ 5 $	$\Q(\sqrt{5}) $	$C_2$ (as 2T1)	$1$	$1$
*	2.2495.4t3.c.a	$2$	$ 5 \cdot 499 $	4.0.1245005.1	$D_{4}$ (as 4T3)	$1$	$0$
*	2.2495.7t2.a.a	$2$	$ 5 \cdot 499 $	7.1.15531437375.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.2495.14t3.a.a	$2$	$ 5 \cdot 499 $	14.0.120371547919839898421875.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.2495.14t3.a.b	$2$	$ 5 \cdot 499 $	14.0.120371547919839898421875.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.2495.7t2.a.c	$2$	$ 5 \cdot 499 $	7.1.15531437375.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.2495.7t2.a.b	$2$	$ 5 \cdot 499 $	7.1.15531437375.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.2495.14t3.a.c	$2$	$ 5 \cdot 499 $	14.0.120371547919839898421875.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.2495.28t10.a.e	$2$	$ 5 \cdot 499 $	28.2.725917312055025700242151478457681268310546875.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.2495.28t10.a.d	$2$	$ 5 \cdot 499 $	28.2.725917312055025700242151478457681268310546875.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.2495.28t10.a.b	$2$	$ 5 \cdot 499 $	28.2.725917312055025700242151478457681268310546875.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.2495.28t10.a.c	$2$	$ 5 \cdot 499 $	28.2.725917312055025700242151478457681268310546875.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.2495.28t10.a.f	$2$	$ 5 \cdot 499 $	28.2.725917312055025700242151478457681268310546875.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.2495.28t10.a.a	$2$	$ 5 \cdot 499 $	28.2.725917312055025700242151478457681268310546875.1	$D_{28}$ (as 28T10)	$1$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

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