Normalized defining polynomial

\( x^{27} - 36 x^{24} + 752 x^{21} + 528 x^{18} - 2616 x^{15} + 44416 x^{12} + 37248 x^{9} + 11712 x^{6} + \cdots + 64 \) x^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64

Invariants

Degree:		$27$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[3, 12]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(3537182715531733726396416000000000000000000\) 3537182715531733726396416000000000000000000 \(\medspace = 2^{52}\cdot 3^{30}\cdot 5^{18}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(37.66\)	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:		$2^{25/12}3^{7/6}5^{3/4}\approx 51.05223872742641$
Ramified primes:		\(2\), \(3\), \(5\) 2, 3, 5	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{24}a^{11}-\frac{1}{24}a^{10}+\frac{1}{24}a^{9}-\frac{5}{12}a^{5}+\frac{5}{12}a^{4}-\frac{5}{12}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{24}a^{12}+\frac{1}{24}a^{9}+\frac{1}{12}a^{6}+\frac{1}{4}a^{3}-\frac{1}{3}$, $\frac{1}{72}a^{13}+\frac{1}{72}a^{12}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}+\frac{1}{36}a^{7}+\frac{1}{36}a^{6}-\frac{1}{9}a-\frac{1}{9}$, $\frac{1}{144}a^{14}+\frac{1}{72}a^{12}-\frac{1}{72}a^{11}+\frac{1}{24}a^{10}+\frac{1}{72}a^{9}+\frac{1}{72}a^{8}+\frac{1}{36}a^{6}-\frac{1}{12}a^{5}-\frac{5}{12}a^{4}+\frac{5}{12}a^{3}+\frac{5}{18}a^{2}-\frac{1}{3}a+\frac{2}{9}$, $\frac{1}{1728}a^{15}-\frac{1}{432}a^{12}-\frac{7}{144}a^{9}+\frac{2}{27}a^{6}-\frac{119}{432}a^{3}+\frac{13}{108}$, $\frac{1}{1728}a^{16}-\frac{1}{432}a^{13}-\frac{7}{144}a^{10}+\frac{2}{27}a^{7}-\frac{119}{432}a^{4}+\frac{13}{108}a$, $\frac{1}{5184}a^{17}-\frac{1}{5184}a^{16}+\frac{1}{5184}a^{15}-\frac{1}{1296}a^{14}+\frac{1}{1296}a^{13}-\frac{1}{1296}a^{12}-\frac{7}{432}a^{11}+\frac{7}{432}a^{10}-\frac{7}{432}a^{9}+\frac{2}{81}a^{8}-\frac{2}{81}a^{7}+\frac{2}{81}a^{6}-\frac{551}{1296}a^{5}+\frac{551}{1296}a^{4}-\frac{551}{1296}a^{3}-\frac{95}{324}a^{2}+\frac{95}{324}a-\frac{95}{324}$, $\frac{1}{5184}a^{18}-\frac{25}{1296}a^{12}-\frac{13}{324}a^{9}+\frac{25}{144}a^{6}+\frac{1}{162}a^{3}-\frac{14}{81}$, $\frac{1}{15552}a^{19}+\frac{1}{15552}a^{18}-\frac{1}{5184}a^{16}-\frac{1}{5184}a^{15}-\frac{11}{1944}a^{13}-\frac{11}{1944}a^{12}+\frac{11}{3888}a^{10}+\frac{11}{3888}a^{9}+\frac{43}{1296}a^{7}+\frac{43}{1296}a^{6}+\frac{1661}{3888}a^{4}+\frac{1661}{3888}a^{3}-\frac{95}{972}a-\frac{95}{972}$, $\frac{1}{15552}a^{20}-\frac{1}{15552}a^{18}-\frac{1}{5184}a^{16}-\frac{1}{5184}a^{15}+\frac{1}{1944}a^{14}+\frac{1}{1296}a^{13}-\frac{5}{243}a^{12}+\frac{7}{486}a^{11}+\frac{7}{432}a^{10}+\frac{169}{3888}a^{9}+\frac{31}{432}a^{8}-\frac{2}{81}a^{7}-\frac{179}{1296}a^{6}-\frac{121}{243}a^{5}+\frac{551}{1296}a^{4}+\frac{673}{3888}a^{3}-\frac{217}{486}a^{2}+\frac{95}{324}a-\frac{91}{972}$, $\frac{1}{186624}a^{21}+\frac{1}{11664}a^{18}-\frac{5}{93312}a^{15}-\frac{271}{23328}a^{12}+\frac{1391}{46656}a^{9}+\frac{325}{5832}a^{6}-\frac{1369}{7776}a^{3}-\frac{2455}{5832}$, $\frac{1}{1119744}a^{22}-\frac{1}{559872}a^{21}-\frac{5}{279936}a^{19}+\frac{5}{139968}a^{18}+\frac{49}{559872}a^{16}-\frac{49}{279936}a^{15}+\frac{125}{139968}a^{13}-\frac{125}{69984}a^{12}-\frac{16501}{279936}a^{10}-\frac{995}{139968}a^{9}+\frac{11153}{69984}a^{7}+\frac{6343}{34992}a^{6}+\frac{10049}{46656}a^{4}-\frac{4217}{23328}a^{3}-\frac{12409}{34992}a-\frac{5087}{17496}$, $\frac{1}{1119744}a^{23}+\frac{1}{559872}a^{21}-\frac{5}{279936}a^{20}-\frac{5}{139968}a^{18}+\frac{49}{559872}a^{17}+\frac{49}{279936}a^{15}+\frac{125}{139968}a^{14}+\frac{125}{69984}a^{12}-\frac{4837}{279936}a^{11}-\frac{1}{24}a^{10}+\frac{6827}{139968}a^{9}-\frac{6343}{69984}a^{8}-\frac{6343}{34992}a^{6}-\frac{9391}{46656}a^{5}+\frac{5}{12}a^{4}-\frac{5503}{23328}a^{3}-\frac{6577}{34992}a^{2}+\frac{1}{3}a-\frac{745}{17496}$, $\frac{1}{3359232}a^{24}+\frac{1}{419904}a^{21}+\frac{31}{559872}a^{18}-\frac{1}{23328}a^{15}+\frac{4393}{279936}a^{12}+\frac{6305}{104976}a^{9}-\frac{69425}{419904}a^{6}+\frac{23575}{52488}a^{3}-\frac{10237}{26244}$, $\frac{1}{3359232}a^{25}-\frac{1}{3359232}a^{22}-\frac{11}{559872}a^{19}+\frac{1}{15552}a^{18}+\frac{5}{62208}a^{16}-\frac{1}{5184}a^{15}-\frac{965}{279936}a^{13}+\frac{2}{243}a^{12}-\frac{4091}{839808}a^{10}+\frac{227}{3888}a^{9}+\frac{98533}{419904}a^{7}+\frac{79}{1296}a^{6}+\frac{83381}{419904}a^{4}+\frac{1661}{3888}a^{3}+\frac{9605}{104976}a-\frac{203}{972}$, $\frac{1}{10077696}a^{26}-\frac{1}{10077696}a^{25}+\frac{1}{10077696}a^{24}-\frac{1}{10077696}a^{23}+\frac{1}{10077696}a^{22}+\frac{13}{5038848}a^{21}-\frac{47}{1679616}a^{20}+\frac{47}{1679616}a^{19}-\frac{137}{1679616}a^{18}+\frac{17}{186624}a^{17}-\frac{17}{186624}a^{16}-\frac{7}{31104}a^{15}+\frac{2563}{839808}a^{14}+\frac{3269}{839808}a^{13}+\frac{12589}{839808}a^{12}+\frac{16861}{2519424}a^{11}+\frac{53123}{2519424}a^{10}+\frac{39359}{1259712}a^{9}-\frac{119519}{1259712}a^{8}+\frac{137015}{1259712}a^{7}-\frac{222929}{1259712}a^{6}-\frac{305959}{1259712}a^{5}-\frac{323897}{1259712}a^{4}-\frac{44885}{629856}a^{3}+\frac{119009}{314928}a^{2}-\frac{136505}{314928}a+\frac{21745}{157464}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6, 1/2*a^7, 1/4*a^8 - 1/2*a^2, 1/8*a^9 - 1/4*a^3, 1/8*a^10 - 1/4*a^4, 1/24*a^11 - 1/24*a^10 + 1/24*a^9 - 5/12*a^5 + 5/12*a^4 - 5/12*a^3 - 1/3*a^2 + 1/3*a - 1/3, 1/24*a^12 + 1/24*a^9 + 1/12*a^6 + 1/4*a^3 - 1/3, 1/72*a^13 + 1/72*a^12 + 1/18*a^10 + 1/18*a^9 + 1/36*a^7 + 1/36*a^6 - 1/9*a - 1/9, 1/144*a^14 + 1/72*a^12 - 1/72*a^11 + 1/24*a^10 + 1/72*a^9 + 1/72*a^8 + 1/36*a^6 - 1/12*a^5 - 5/12*a^4 + 5/12*a^3 + 5/18*a^2 - 1/3*a + 2/9, 1/1728*a^15 - 1/432*a^12 - 7/144*a^9 + 2/27*a^6 - 119/432*a^3 + 13/108, 1/1728*a^16 - 1/432*a^13 - 7/144*a^10 + 2/27*a^7 - 119/432*a^4 + 13/108*a, 1/5184*a^17 - 1/5184*a^16 + 1/5184*a^15 - 1/1296*a^14 + 1/1296*a^13 - 1/1296*a^12 - 7/432*a^11 + 7/432*a^10 - 7/432*a^9 + 2/81*a^8 - 2/81*a^7 + 2/81*a^6 - 551/1296*a^5 + 551/1296*a^4 - 551/1296*a^3 - 95/324*a^2 + 95/324*a - 95/324, 1/5184*a^18 - 25/1296*a^12 - 13/324*a^9 + 25/144*a^6 + 1/162*a^3 - 14/81, 1/15552*a^19 + 1/15552*a^18 - 1/5184*a^16 - 1/5184*a^15 - 11/1944*a^13 - 11/1944*a^12 + 11/3888*a^10 + 11/3888*a^9 + 43/1296*a^7 + 43/1296*a^6 + 1661/3888*a^4 + 1661/3888*a^3 - 95/972*a - 95/972, 1/15552*a^20 - 1/15552*a^18 - 1/5184*a^16 - 1/5184*a^15 + 1/1944*a^14 + 1/1296*a^13 - 5/243*a^12 + 7/486*a^11 + 7/432*a^10 + 169/3888*a^9 + 31/432*a^8 - 2/81*a^7 - 179/1296*a^6 - 121/243*a^5 + 551/1296*a^4 + 673/3888*a^3 - 217/486*a^2 + 95/324*a - 91/972, 1/186624*a^21 + 1/11664*a^18 - 5/93312*a^15 - 271/23328*a^12 + 1391/46656*a^9 + 325/5832*a^6 - 1369/7776*a^3 - 2455/5832, 1/1119744*a^22 - 1/559872*a^21 - 5/279936*a^19 + 5/139968*a^18 + 49/559872*a^16 - 49/279936*a^15 + 125/139968*a^13 - 125/69984*a^12 - 16501/279936*a^10 - 995/139968*a^9 + 11153/69984*a^7 + 6343/34992*a^6 + 10049/46656*a^4 - 4217/23328*a^3 - 12409/34992*a - 5087/17496, 1/1119744*a^23 + 1/559872*a^21 - 5/279936*a^20 - 5/139968*a^18 + 49/559872*a^17 + 49/279936*a^15 + 125/139968*a^14 + 125/69984*a^12 - 4837/279936*a^11 - 1/24*a^10 + 6827/139968*a^9 - 6343/69984*a^8 - 6343/34992*a^6 - 9391/46656*a^5 + 5/12*a^4 - 5503/23328*a^3 - 6577/34992*a^2 + 1/3*a - 745/17496, 1/3359232*a^24 + 1/419904*a^21 + 31/559872*a^18 - 1/23328*a^15 + 4393/279936*a^12 + 6305/104976*a^9 - 69425/419904*a^6 + 23575/52488*a^3 - 10237/26244, 1/3359232*a^25 - 1/3359232*a^22 - 11/559872*a^19 + 1/15552*a^18 + 5/62208*a^16 - 1/5184*a^15 - 965/279936*a^13 + 2/243*a^12 - 4091/839808*a^10 + 227/3888*a^9 + 98533/419904*a^7 + 79/1296*a^6 + 83381/419904*a^4 + 1661/3888*a^3 + 9605/104976*a - 203/972, 1/10077696*a^26 - 1/10077696*a^25 + 1/10077696*a^24 - 1/10077696*a^23 + 1/10077696*a^22 + 13/5038848*a^21 - 47/1679616*a^20 + 47/1679616*a^19 - 137/1679616*a^18 + 17/186624*a^17 - 17/186624*a^16 - 7/31104*a^15 + 2563/839808*a^14 + 3269/839808*a^13 + 12589/839808*a^12 + 16861/2519424*a^11 + 53123/2519424*a^10 + 39359/1259712*a^9 - 119519/1259712*a^8 + 137015/1259712*a^7 - 222929/1259712*a^6 - 305959/1259712*a^5 - 323897/1259712*a^4 - 44885/629856*a^3 + 119009/314928*a^2 - 136505/314928*a + 21745/157464

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

Unit group

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:		$\frac{27049}{1259712}a^{26}-\frac{43979}{10077696}a^{25}+\frac{61997}{10077696}a^{24}-\frac{7816073}{10077696}a^{23}+\frac{1594193}{10077696}a^{22}-\frac{1124431}{5038848}a^{21}+\frac{13638859}{839808}a^{20}-\frac{5578319}{1679616}a^{19}+\frac{7873055}{1679616}a^{18}+\frac{5252971}{559872}a^{17}-\frac{825467}{559872}a^{16}+\frac{61009}{31104}a^{15}-\frac{24001307}{419904}a^{14}+\frac{9843115}{839808}a^{13}-\frac{13900591}{839808}a^{12}+\frac{2420173685}{2519424}a^{11}-\frac{495711101}{2519424}a^{10}+\frac{349928443}{1259712}a^{9}+\frac{430869247}{629856}a^{8}-\frac{142806839}{1259712}a^{7}+\frac{192578495}{1259712}a^{6}+\frac{223762705}{1259712}a^{5}-\frac{33368065}{1259712}a^{4}+\frac{21967091}{629856}a^{3}+\frac{5227309}{314928}a^{2}-\frac{404281}{314928}a+\frac{345353}{157464}$, $\frac{3757}{279936}a^{26}+\frac{475}{62208}a^{25}-\frac{7303}{1119744}a^{24}+\frac{545017}{1119744}a^{23}-\frac{154231}{559872}a^{22}+\frac{16523}{69984}a^{21}-\frac{105989}{10368}a^{20}+\frac{1613383}{279936}a^{19}-\frac{2772379}{559872}a^{18}-\frac{811333}{186624}a^{17}+\frac{1003229}{279936}a^{16}-\frac{42901}{17496}a^{15}+\frac{1680677}{46656}a^{14}-\frac{2829865}{139968}a^{13}+\frac{4903795}{279936}a^{12}-\frac{169593181}{279936}a^{11}+\frac{47686771}{139968}a^{10}-\frac{641218}{2187}a^{9}-\frac{23612189}{69984}a^{8}+\frac{18051137}{69984}a^{7}-\frac{8593363}{46656}a^{6}-\frac{11435021}{139968}a^{5}+\frac{1667629}{23328}a^{4}-\frac{362693}{8748}a^{3}-\frac{299153}{34992}a^{2}+\frac{101767}{17496}a-\frac{7429}{2916}$, $\frac{1315}{209952}a^{26}-\frac{1315}{209952}a^{25}+\frac{1315}{209952}a^{24}-\frac{189901}{839808}a^{23}+\frac{189901}{839808}a^{22}-\frac{189901}{839808}a^{21}+\frac{220829}{46656}a^{20}-\frac{220829}{46656}a^{19}+\frac{220829}{46656}a^{18}+\frac{197765}{69984}a^{17}-\frac{197765}{69984}a^{16}+\frac{197765}{69984}a^{15}-\frac{587663}{34992}a^{14}+\frac{587663}{34992}a^{13}-\frac{587663}{34992}a^{12}+\frac{58763875}{209952}a^{11}-\frac{58763875}{209952}a^{10}+\frac{58763875}{209952}a^{9}+\frac{21516001}{104976}a^{8}-\frac{21516001}{104976}a^{7}+\frac{21516001}{104976}a^{6}+\frac{1181363}{26244}a^{5}-\frac{1181363}{26244}a^{4}+\frac{1181363}{26244}a^{3}+\frac{14615}{6561}a^{2}-\frac{14615}{6561}a+\frac{14615}{6561}$, $\frac{9161}{1679616}a^{24}+\frac{167611}{839808}a^{21}-\frac{1180967}{279936}a^{18}-\frac{9809}{23328}a^{15}+\frac{2147815}{139968}a^{12}-\frac{52707229}{209952}a^{9}-\frac{12056111}{209952}a^{6}+\frac{506137}{26244}a^{3}+\frac{47039}{13122}$, $\frac{3863}{629856}a^{26}-\frac{23743}{2519424}a^{25}-\frac{26993}{10077696}a^{24}+\frac{2217443}{10077696}a^{23}+\frac{858205}{2519424}a^{22}+\frac{121255}{1259712}a^{21}-\frac{3849931}{839808}a^{20}-\frac{2996665}{419904}a^{19}-\frac{3372671}{1679616}a^{18}-\frac{2142965}{559872}a^{17}-\frac{183473}{46656}a^{16}-\frac{54103}{34992}a^{15}+\frac{6735053}{419904}a^{14}+\frac{5277143}{209952}a^{13}+\frac{5888791}{839808}a^{12}-\frac{680939615}{2519424}a^{11}-\frac{265966465}{629856}a^{10}-\frac{37329391}{314928}a^{9}-\frac{165999787}{629856}a^{8}-\frac{91076689}{314928}a^{7}-\frac{135573743}{1259712}a^{6}-\frac{96274087}{1259712}a^{5}-\frac{23248481}{314928}a^{4}-\frac{2541361}{78732}a^{3}-\frac{2118691}{314928}a^{2}-\frac{621713}{78732}a-\frac{245953}{78732}$, $\frac{4717}{186624}a^{26}+\frac{17}{3888}a^{25}+\frac{17405}{1679616}a^{24}-\frac{512939}{559872}a^{23}-\frac{3727}{23328}a^{22}-\frac{629903}{1679616}a^{21}+\frac{5384213}{279936}a^{20}+\frac{39355}{11664}a^{19}+\frac{733813}{93312}a^{18}+\frac{2404867}{279936}a^{17}+\frac{23099}{46656}a^{16}+\frac{1112477}{279936}a^{15}-\frac{9571673}{139968}a^{14}-\frac{137969}{11664}a^{13}-\frac{3912485}{139968}a^{12}+\frac{159493931}{139968}a^{11}+\frac{2339999}{11664}a^{10}+\frac{195493463}{419904}a^{9}+\frac{46181275}{69984}a^{8}+\frac{80371}{1458}a^{7}+\frac{62429747}{209952}a^{6}+\frac{2905951}{23328}a^{5}+\frac{18251}{1296}a^{4}+\frac{12493345}{209952}a^{3}+\frac{93257}{17496}a^{2}+\frac{3559}{2916}a+\frac{144433}{52488}$, $\frac{6545}{279936}a^{26}+\frac{8015}{839808}a^{25}-\frac{34325}{3359232}a^{24}+\frac{471889}{559872}a^{23}-\frac{1159547}{3359232}a^{22}+\frac{77843}{209952}a^{21}-\frac{91361}{5184}a^{20}+\frac{675125}{93312}a^{19}-\frac{4361275}{559872}a^{18}-\frac{1070917}{93312}a^{17}+\frac{2137355}{559872}a^{16}-\frac{222053}{69984}a^{15}+\frac{90179}{1458}a^{14}-\frac{3567091}{139968}a^{13}+\frac{858971}{31104}a^{12}-\frac{145771141}{139968}a^{11}+\frac{359634719}{839808}a^{10}-\frac{6059225}{13122}a^{9}-\frac{28680911}{34992}a^{8}+\frac{59425663}{209952}a^{7}-\frac{104475323}{419904}a^{6}-\frac{15835997}{69984}a^{5}+\frac{30045691}{419904}a^{4}-\frac{2619959}{52488}a^{3}-\frac{374261}{17496}a^{2}+\frac{720943}{104976}a-\frac{92071}{26244}$, $\frac{45161}{5038848}a^{26}-\frac{26381}{10077696}a^{25}+\frac{16619}{10077696}a^{24}+\frac{3276349}{10077696}a^{23}+\frac{474535}{5038848}a^{22}-\frac{299995}{5038848}a^{21}-\frac{1433705}{209952}a^{20}-\frac{3302255}{1679616}a^{19}+\frac{2093033}{1679616}a^{18}-\frac{178471}{62208}a^{17}-\frac{402167}{279936}a^{16}+\frac{208831}{279936}a^{15}+\frac{5124905}{209952}a^{14}+\frac{5833639}{839808}a^{13}-\frac{3743977}{839808}a^{12}-\frac{1019626201}{2519424}a^{11}-\frac{146233831}{1259712}a^{10}+\frac{92833723}{1259712}a^{9}-\frac{17599789}{78732}a^{8}-\frac{126916607}{1259712}a^{7}+\frac{68190089}{1259712}a^{6}-\frac{41828813}{1259712}a^{5}-\frac{16020401}{629856}a^{4}+\frac{6093449}{629856}a^{3}-\frac{52601}{314928}a^{2}-\frac{321383}{157464}a+\frac{88259}{157464}$, $\frac{601}{93312}a^{26}-\frac{4469}{124416}a^{25}-\frac{25427}{1679616}a^{24}+\frac{129889}{559872}a^{23}+\frac{30359}{23328}a^{22}+\frac{461479}{839808}a^{21}-\frac{678467}{139968}a^{20}-\frac{5097107}{186624}a^{19}-\frac{3232831}{279936}a^{18}-\frac{933167}{279936}a^{17}-\frac{295205}{23328}a^{16}-\frac{79369}{17496}a^{15}+\frac{153697}{8748}a^{14}+\frac{9016379}{93312}a^{13}+\frac{1899683}{46656}a^{12}-\frac{40078165}{139968}a^{11}-\frac{9434603}{5832}a^{10}-\frac{143732977}{209952}a^{9}-\frac{8273245}{34992}a^{8}-\frac{44973505}{46656}a^{7}-\frac{75336827}{209952}a^{6}-\frac{742175}{23328}a^{5}-\frac{415609}{1944}a^{4}-\frac{4424695}{52488}a^{3}+\frac{12443}{17496}a^{2}-\frac{36821}{2916}a-\frac{33335}{6561}$, $\frac{5771}{10077696}a^{26}-\frac{34999}{10077696}a^{25}+\frac{5089}{10077696}a^{24}+\frac{104491}{5038848}a^{23}+\frac{319963}{2519424}a^{22}-\frac{93611}{5038848}a^{21}-\frac{730889}{1679616}a^{20}-\frac{4506769}{1679616}a^{19}+\frac{662335}{1679616}a^{18}-\frac{57619}{279936}a^{17}-\frac{23117}{69984}a^{16}-\frac{11743}{279936}a^{15}+\frac{1223233}{839808}a^{14}+\frac{8164865}{839808}a^{13}-\frac{1131971}{839808}a^{12}-\frac{32513251}{1259712}a^{11}-\frac{100558429}{629856}a^{10}+\frac{29675807}{1259712}a^{9}-\frac{19568225}{1259712}a^{8}-\frac{50908417}{1259712}a^{7}+\frac{529447}{1259712}a^{6}-\frac{5180189}{629856}a^{5}+\frac{685975}{78732}a^{4}+\frac{887779}{629856}a^{3}-\frac{449039}{157464}a^{2}+\frac{185437}{78732}a+\frac{137353}{157464}$, $\frac{62149}{5038848}a^{26}-\frac{123581}{10077696}a^{25}-\frac{3889}{2519424}a^{24}-\frac{2228323}{5038848}a^{23}+\frac{2235421}{5038848}a^{22}+\frac{140077}{2519424}a^{21}+\frac{7734139}{839808}a^{20}-\frac{15620915}{1679616}a^{19}-\frac{7621}{6561}a^{18}+\frac{2212327}{279936}a^{17}-\frac{1351277}{279936}a^{16}-\frac{37477}{46656}a^{15}-\frac{13515737}{419904}a^{14}+\frac{27703063}{839808}a^{13}+\frac{895439}{209952}a^{12}+\frac{683777059}{1259712}a^{11}-\frac{693531661}{1259712}a^{10}-\frac{43206817}{629856}a^{9}+\frac{341518963}{629856}a^{8}-\frac{452598923}{1259712}a^{7}-\frac{8970269}{157464}a^{6}+\frac{100566497}{629856}a^{5}-\frac{48281243}{629856}a^{4}-\frac{1373333}{314928}a^{3}+\frac{2272913}{157464}a^{2}-\frac{1166969}{157464}a+\frac{110923}{78732}$, $\frac{7927}{10077696}a^{26}+\frac{63691}{5038848}a^{25}-\frac{49693}{5038848}a^{24}-\frac{171089}{5038848}a^{23}-\frac{4601093}{10077696}a^{22}+\frac{225635}{629856}a^{21}+\frac{1338625}{1679616}a^{20}+\frac{4014515}{419904}a^{19}-\frac{6325825}{839808}a^{18}-\frac{1096243}{279936}a^{17}+\frac{1028629}{186624}a^{16}-\frac{386743}{139968}a^{15}-\frac{2592893}{839808}a^{14}-\frac{879275}{26244}a^{13}+\frac{11220635}{419904}a^{12}+\frac{63378185}{1259712}a^{11}+\frac{1424774369}{2519424}a^{10}-\frac{140673109}{314928}a^{9}-\frac{287167943}{1259712}a^{8}+\frac{126583577}{314928}a^{7}-\frac{140094157}{629856}a^{6}-\frac{53678957}{629856}a^{5}+\frac{142365229}{1259712}a^{4}-\frac{13164787}{314928}a^{3}-\frac{2008631}{157464}a^{2}+\frac{3757537}{314928}a-\frac{156289}{78732}$, $\frac{5173}{10077696}a^{26}-\frac{39409}{10077696}a^{25}-\frac{12359}{10077696}a^{24}-\frac{159577}{10077696}a^{23}+\frac{709577}{5038848}a^{22}+\frac{56599}{1259712}a^{21}+\frac{487009}{1679616}a^{20}-\frac{4941811}{1679616}a^{19}-\frac{1596581}{1679616}a^{18}+\frac{1282507}{559872}a^{17}-\frac{569201}{279936}a^{16}-\frac{7325}{139968}a^{15}-\frac{501545}{839808}a^{14}+\frac{8621867}{839808}a^{13}+\frac{2898817}{839808}a^{12}+\frac{39320245}{2519424}a^{11}-\frac{218940305}{1259712}a^{10}-\frac{1114888}{19683}a^{9}+\frac{175118929}{1259712}a^{8}-\frac{181094947}{1259712}a^{7}-\frac{13445165}{1259712}a^{6}+\frac{83038337}{1259712}a^{5}-\frac{26838175}{629856}a^{4}+\frac{1286471}{314928}a^{3}+\frac{2741513}{314928}a^{2}-\frac{662533}{157464}a+\frac{20783}{19683}$, $\frac{64615}{2519424}a^{26}-\frac{28361}{1259712}a^{25}+\frac{25393}{2519424}a^{24}-\frac{2336899}{2519424}a^{23}+\frac{4110155}{5038848}a^{22}-\frac{229987}{629856}a^{21}+\frac{8163259}{419904}a^{20}-\frac{3594095}{209952}a^{19}+\frac{402203}{52488}a^{18}+\frac{1442377}{139968}a^{17}-\frac{2221775}{279936}a^{16}+\frac{499717}{139968}a^{15}-\frac{14449307}{209952}a^{14}+\frac{12743663}{209952}a^{13}-\frac{5701703}{209952}a^{12}+\frac{724721731}{629856}a^{11}-\frac{1277357243}{1259712}a^{10}+\frac{142941833}{314928}a^{9}+\frac{240523987}{314928}a^{8}-\frac{47597743}{78732}a^{7}+\frac{42778721}{157464}a^{6}+\frac{54263999}{314928}a^{5}-\frac{79375657}{629856}a^{4}+\frac{18030989}{314928}a^{3}+\frac{1126835}{78732}a^{2}-\frac{1437397}{157464}a+\frac{314045}{78732}$ 27049/1259712*a^26 - 43979/10077696*a^25 + 61997/10077696*a^24 - 7816073/10077696*a^23 + 1594193/10077696*a^22 - 1124431/5038848*a^21 + 13638859/839808*a^20 - 5578319/1679616*a^19 + 7873055/1679616*a^18 + 5252971/559872*a^17 - 825467/559872*a^16 + 61009/31104*a^15 - 24001307/419904*a^14 + 9843115/839808*a^13 - 13900591/839808*a^12 + 2420173685/2519424*a^11 - 495711101/2519424*a^10 + 349928443/1259712*a^9 + 430869247/629856*a^8 - 142806839/1259712*a^7 + 192578495/1259712*a^6 + 223762705/1259712*a^5 - 33368065/1259712*a^4 + 21967091/629856*a^3 + 5227309/314928*a^2 - 404281/314928*a + 345353/157464, -3757/279936*a^26 + 475/62208*a^25 - 7303/1119744*a^24 + 545017/1119744*a^23 - 154231/559872*a^22 + 16523/69984*a^21 - 105989/10368*a^20 + 1613383/279936*a^19 - 2772379/559872*a^18 - 811333/186624*a^17 + 1003229/279936*a^16 - 42901/17496*a^15 + 1680677/46656*a^14 - 2829865/139968*a^13 + 4903795/279936*a^12 - 169593181/279936*a^11 + 47686771/139968*a^10 - 641218/2187*a^9 - 23612189/69984*a^8 + 18051137/69984*a^7 - 8593363/46656*a^6 - 11435021/139968*a^5 + 1667629/23328*a^4 - 362693/8748*a^3 - 299153/34992*a^2 + 101767/17496*a - 7429/2916, 1315/209952*a^26 - 1315/209952*a^25 + 1315/209952*a^24 - 189901/839808*a^23 + 189901/839808*a^22 - 189901/839808*a^21 + 220829/46656*a^20 - 220829/46656*a^19 + 220829/46656*a^18 + 197765/69984*a^17 - 197765/69984*a^16 + 197765/69984*a^15 - 587663/34992*a^14 + 587663/34992*a^13 - 587663/34992*a^12 + 58763875/209952*a^11 - 58763875/209952*a^10 + 58763875/209952*a^9 + 21516001/104976*a^8 - 21516001/104976*a^7 + 21516001/104976*a^6 + 1181363/26244*a^5 - 1181363/26244*a^4 + 1181363/26244*a^3 + 14615/6561*a^2 - 14615/6561*a + 14615/6561, -9161/1679616*a^24 + 167611/839808*a^21 - 1180967/279936*a^18 - 9809/23328*a^15 + 2147815/139968*a^12 - 52707229/209952*a^9 - 12056111/209952*a^6 + 506137/26244*a^3 + 47039/13122, -3863/629856*a^26 - 23743/2519424*a^25 - 26993/10077696*a^24 + 2217443/10077696*a^23 + 858205/2519424*a^22 + 121255/1259712*a^21 - 3849931/839808*a^20 - 2996665/419904*a^19 - 3372671/1679616*a^18 - 2142965/559872*a^17 - 183473/46656*a^16 - 54103/34992*a^15 + 6735053/419904*a^14 + 5277143/209952*a^13 + 5888791/839808*a^12 - 680939615/2519424*a^11 - 265966465/629856*a^10 - 37329391/314928*a^9 - 165999787/629856*a^8 - 91076689/314928*a^7 - 135573743/1259712*a^6 - 96274087/1259712*a^5 - 23248481/314928*a^4 - 2541361/78732*a^3 - 2118691/314928*a^2 - 621713/78732*a - 245953/78732, 4717/186624*a^26 + 17/3888*a^25 + 17405/1679616*a^24 - 512939/559872*a^23 - 3727/23328*a^22 - 629903/1679616*a^21 + 5384213/279936*a^20 + 39355/11664*a^19 + 733813/93312*a^18 + 2404867/279936*a^17 + 23099/46656*a^16 + 1112477/279936*a^15 - 9571673/139968*a^14 - 137969/11664*a^13 - 3912485/139968*a^12 + 159493931/139968*a^11 + 2339999/11664*a^10 + 195493463/419904*a^9 + 46181275/69984*a^8 + 80371/1458*a^7 + 62429747/209952*a^6 + 2905951/23328*a^5 + 18251/1296*a^4 + 12493345/209952*a^3 + 93257/17496*a^2 + 3559/2916*a + 144433/52488, -6545/279936*a^26 + 8015/839808*a^25 - 34325/3359232*a^24 + 471889/559872*a^23 - 1159547/3359232*a^22 + 77843/209952*a^21 - 91361/5184*a^20 + 675125/93312*a^19 - 4361275/559872*a^18 - 1070917/93312*a^17 + 2137355/559872*a^16 - 222053/69984*a^15 + 90179/1458*a^14 - 3567091/139968*a^13 + 858971/31104*a^12 - 145771141/139968*a^11 + 359634719/839808*a^10 - 6059225/13122*a^9 - 28680911/34992*a^8 + 59425663/209952*a^7 - 104475323/419904*a^6 - 15835997/69984*a^5 + 30045691/419904*a^4 - 2619959/52488*a^3 - 374261/17496*a^2 + 720943/104976*a - 92071/26244, -45161/5038848*a^26 - 26381/10077696*a^25 + 16619/10077696*a^24 + 3276349/10077696*a^23 + 474535/5038848*a^22 - 299995/5038848*a^21 - 1433705/209952*a^20 - 3302255/1679616*a^19 + 2093033/1679616*a^18 - 178471/62208*a^17 - 402167/279936*a^16 + 208831/279936*a^15 + 5124905/209952*a^14 + 5833639/839808*a^13 - 3743977/839808*a^12 - 1019626201/2519424*a^11 - 146233831/1259712*a^10 + 92833723/1259712*a^9 - 17599789/78732*a^8 - 126916607/1259712*a^7 + 68190089/1259712*a^6 - 41828813/1259712*a^5 - 16020401/629856*a^4 + 6093449/629856*a^3 - 52601/314928*a^2 - 321383/157464*a + 88259/157464, -601/93312*a^26 - 4469/124416*a^25 - 25427/1679616*a^24 + 129889/559872*a^23 + 30359/23328*a^22 + 461479/839808*a^21 - 678467/139968*a^20 - 5097107/186624*a^19 - 3232831/279936*a^18 - 933167/279936*a^17 - 295205/23328*a^16 - 79369/17496*a^15 + 153697/8748*a^14 + 9016379/93312*a^13 + 1899683/46656*a^12 - 40078165/139968*a^11 - 9434603/5832*a^10 - 143732977/209952*a^9 - 8273245/34992*a^8 - 44973505/46656*a^7 - 75336827/209952*a^6 - 742175/23328*a^5 - 415609/1944*a^4 - 4424695/52488*a^3 + 12443/17496*a^2 - 36821/2916*a - 33335/6561, -5771/10077696*a^26 - 34999/10077696*a^25 + 5089/10077696*a^24 + 104491/5038848*a^23 + 319963/2519424*a^22 - 93611/5038848*a^21 - 730889/1679616*a^20 - 4506769/1679616*a^19 + 662335/1679616*a^18 - 57619/279936*a^17 - 23117/69984*a^16 - 11743/279936*a^15 + 1223233/839808*a^14 + 8164865/839808*a^13 - 1131971/839808*a^12 - 32513251/1259712*a^11 - 100558429/629856*a^10 + 29675807/1259712*a^9 - 19568225/1259712*a^8 - 50908417/1259712*a^7 + 529447/1259712*a^6 - 5180189/629856*a^5 + 685975/78732*a^4 + 887779/629856*a^3 - 449039/157464*a^2 + 185437/78732*a + 137353/157464, 62149/5038848*a^26 - 123581/10077696*a^25 - 3889/2519424*a^24 - 2228323/5038848*a^23 + 2235421/5038848*a^22 + 140077/2519424*a^21 + 7734139/839808*a^20 - 15620915/1679616*a^19 - 7621/6561*a^18 + 2212327/279936*a^17 - 1351277/279936*a^16 - 37477/46656*a^15 - 13515737/419904*a^14 + 27703063/839808*a^13 + 895439/209952*a^12 + 683777059/1259712*a^11 - 693531661/1259712*a^10 - 43206817/629856*a^9 + 341518963/629856*a^8 - 452598923/1259712*a^7 - 8970269/157464*a^6 + 100566497/629856*a^5 - 48281243/629856*a^4 - 1373333/314928*a^3 + 2272913/157464*a^2 - 1166969/157464*a + 110923/78732, 7927/10077696*a^26 + 63691/5038848*a^25 - 49693/5038848*a^24 - 171089/5038848*a^23 - 4601093/10077696*a^22 + 225635/629856*a^21 + 1338625/1679616*a^20 + 4014515/419904*a^19 - 6325825/839808*a^18 - 1096243/279936*a^17 + 1028629/186624*a^16 - 386743/139968*a^15 - 2592893/839808*a^14 - 879275/26244*a^13 + 11220635/419904*a^12 + 63378185/1259712*a^11 + 1424774369/2519424*a^10 - 140673109/314928*a^9 - 287167943/1259712*a^8 + 126583577/314928*a^7 - 140094157/629856*a^6 - 53678957/629856*a^5 + 142365229/1259712*a^4 - 13164787/314928*a^3 - 2008631/157464*a^2 + 3757537/314928*a - 156289/78732, 5173/10077696*a^26 - 39409/10077696*a^25 - 12359/10077696*a^24 - 159577/10077696*a^23 + 709577/5038848*a^22 + 56599/1259712*a^21 + 487009/1679616*a^20 - 4941811/1679616*a^19 - 1596581/1679616*a^18 + 1282507/559872*a^17 - 569201/279936*a^16 - 7325/139968*a^15 - 501545/839808*a^14 + 8621867/839808*a^13 + 2898817/839808*a^12 + 39320245/2519424*a^11 - 218940305/1259712*a^10 - 1114888/19683*a^9 + 175118929/1259712*a^8 - 181094947/1259712*a^7 - 13445165/1259712*a^6 + 83038337/1259712*a^5 - 26838175/629856*a^4 + 1286471/314928*a^3 + 2741513/314928*a^2 - 662533/157464*a + 20783/19683, 64615/2519424*a^26 - 28361/1259712*a^25 + 25393/2519424*a^24 - 2336899/2519424*a^23 + 4110155/5038848*a^22 - 229987/629856*a^21 + 8163259/419904*a^20 - 3594095/209952*a^19 + 402203/52488*a^18 + 1442377/139968*a^17 - 2221775/279936*a^16 + 499717/139968*a^15 - 14449307/209952*a^14 + 12743663/209952*a^13 - 5701703/209952*a^12 + 724721731/629856*a^11 - 1277357243/1259712*a^10 + 142941833/314928*a^9 + 240523987/314928*a^8 - 47597743/78732*a^7 + 42778721/157464*a^6 + 54263999/314928*a^5 - 79375657/629856*a^4 + 18030989/314928*a^3 + 1126835/78732*a^2 - 1437397/157464*a + 314045/78732 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 54642144375.730865 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 54642144375.730865 \cdot 3}{2\cdot\sqrt{3537182715531733726396416000000000000000000}}\cr\approx \mathstrut & 1.31989278152437 \end{aligned}\] (assuming GRH)

Galois group

$\He_3:\GL(2,3)$ (as 27T294):

Intermediate fields

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

Sibling fields

Frobenius cycle types

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

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(2\) 2	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
		Deg $24$	$24$	$1$	$50$
\(3\) 3	3.9.9.6	$x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$	$3$	$3$	$9$	$S_3\times C_3$	$[3/2]_{2}^{3}$
		Deg $18$	$6$	$3$	$21$
\(5\) 5	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.8.6.4	$x^{8} - 20 x^{4} + 50$	$4$	$2$	$6$	$C_8$	$[\ ]_{4}^{2}$
	5.8.6.4	$x^{8} - 20 x^{4} + 50$	$4$	$2$	$6$	$C_8$	$[\ ]_{4}^{2}$
	5.8.6.4	$x^{8} - 20 x^{4} + 50$	$4$	$2$	$6$	$C_8$	$[\ ]_{4}^{2}$