↑

Properties

Label 30.0.607...847.1
Degree $30$
Signature $[0, 15]$
Discriminant $-6.072\times 10^{48}$
Root discriminant \(42.28\)
Ramified primes $17,127$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $D_{30}$ (as 30T14)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849)
 
gp: K = bnfinit(y^30 - 6*y^29 + 38*y^28 - 178*y^27 + 694*y^26 - 2325*y^25 + 6840*y^24 - 17890*y^23 + 42428*y^22 - 92148*y^21 + 185209*y^20 - 348333*y^19 + 617591*y^18 - 1037914*y^17 + 1659175*y^16 - 2519432*y^15 + 3620042*y^14 - 4892313*y^13 + 6168845*y^12 - 7195888*y^11 + 7698034*y^10 - 7475970*y^9 + 6524456*y^8 - 5050948*y^7 + 3416179*y^6 - 1976201*y^5 + 950326*y^4 - 363451*y^3 + 103267*y^2 - 19350*y + 1849, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849)
 

\( x^{30} - 6 x^{29} + 38 x^{28} - 178 x^{27} + 694 x^{26} - 2325 x^{25} + 6840 x^{24} - 17890 x^{23} + \cdots + 1849 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-6072125144349637560639895035214067242224098130847\) \(\medspace = -\,17^{14}\cdot 127^{15}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(42.28\)
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:  $17^{1/2}127^{1/2}\approx 46.46504062195577$
Ramified primes:   \(17\), \(127\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-127}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17}a^{11}-\frac{1}{17}a^{10}+\frac{1}{17}a^{9}+\frac{6}{17}a^{8}+\frac{8}{17}a^{7}-\frac{8}{17}a^{6}-\frac{7}{17}a^{4}-\frac{1}{17}a^{3}+\frac{2}{17}a^{2}-\frac{7}{17}a-\frac{7}{17}$, $\frac{1}{17}a^{12}+\frac{7}{17}a^{9}-\frac{3}{17}a^{8}-\frac{8}{17}a^{6}-\frac{7}{17}a^{5}-\frac{8}{17}a^{4}+\frac{1}{17}a^{3}-\frac{5}{17}a^{2}+\frac{3}{17}a-\frac{7}{17}$, $\frac{1}{17}a^{13}+\frac{7}{17}a^{10}-\frac{3}{17}a^{9}-\frac{8}{17}a^{7}-\frac{7}{17}a^{6}-\frac{8}{17}a^{5}+\frac{1}{17}a^{4}-\frac{5}{17}a^{3}+\frac{3}{17}a^{2}-\frac{7}{17}a$, $\frac{1}{17}a^{14}+\frac{4}{17}a^{10}-\frac{7}{17}a^{9}+\frac{1}{17}a^{8}+\frac{5}{17}a^{7}-\frac{3}{17}a^{6}+\frac{1}{17}a^{5}-\frac{7}{17}a^{4}-\frac{7}{17}a^{3}-\frac{4}{17}a^{2}-\frac{2}{17}a-\frac{2}{17}$, $\frac{1}{17}a^{15}-\frac{3}{17}a^{10}-\frac{3}{17}a^{9}-\frac{2}{17}a^{8}-\frac{1}{17}a^{7}-\frac{1}{17}a^{6}-\frac{7}{17}a^{5}+\frac{4}{17}a^{4}+\frac{7}{17}a^{2}-\frac{8}{17}a-\frac{6}{17}$, $\frac{1}{17}a^{16}-\frac{6}{17}a^{10}+\frac{1}{17}a^{9}+\frac{6}{17}a^{7}+\frac{3}{17}a^{6}+\frac{4}{17}a^{5}-\frac{4}{17}a^{4}+\frac{4}{17}a^{3}-\frac{2}{17}a^{2}+\frac{7}{17}a-\frac{4}{17}$, $\frac{1}{17}a^{17}-\frac{5}{17}a^{10}+\frac{6}{17}a^{9}+\frac{8}{17}a^{8}+\frac{7}{17}a^{6}-\frac{4}{17}a^{5}-\frac{4}{17}a^{4}-\frac{8}{17}a^{3}+\frac{2}{17}a^{2}+\frac{5}{17}a-\frac{8}{17}$, $\frac{1}{17}a^{18}+\frac{1}{17}a^{10}-\frac{4}{17}a^{9}-\frac{4}{17}a^{8}-\frac{4}{17}a^{7}+\frac{7}{17}a^{6}-\frac{4}{17}a^{5}+\frac{8}{17}a^{4}-\frac{3}{17}a^{3}-\frac{2}{17}a^{2}+\frac{8}{17}a-\frac{1}{17}$, $\frac{1}{17}a^{19}-\frac{3}{17}a^{10}-\frac{5}{17}a^{9}+\frac{7}{17}a^{8}-\frac{1}{17}a^{7}+\frac{4}{17}a^{6}+\frac{8}{17}a^{5}+\frac{4}{17}a^{4}-\frac{1}{17}a^{3}+\frac{6}{17}a^{2}+\frac{6}{17}a+\frac{7}{17}$, $\frac{1}{17}a^{20}-\frac{8}{17}a^{10}-\frac{7}{17}a^{9}-\frac{6}{17}a^{7}+\frac{1}{17}a^{6}+\frac{4}{17}a^{5}-\frac{5}{17}a^{4}+\frac{3}{17}a^{3}-\frac{5}{17}a^{2}+\frac{3}{17}a-\frac{4}{17}$, $\frac{1}{187}a^{21}+\frac{2}{187}a^{20}-\frac{3}{187}a^{19}-\frac{3}{187}a^{18}-\frac{3}{187}a^{17}-\frac{3}{187}a^{16}+\frac{3}{187}a^{15}+\frac{1}{187}a^{13}+\frac{1}{187}a^{12}+\frac{2}{187}a^{11}+\frac{4}{187}a^{10}-\frac{54}{187}a^{9}+\frac{63}{187}a^{8}+\frac{38}{187}a^{7}-\frac{19}{187}a^{6}+\frac{23}{187}a^{5}+\frac{69}{187}a^{4}+\frac{28}{187}a^{3}+\frac{37}{187}a^{2}-\frac{4}{187}a+\frac{4}{11}$, $\frac{1}{3179}a^{22}-\frac{2}{3179}a^{21}+\frac{8}{289}a^{20}-\frac{24}{3179}a^{19}-\frac{46}{3179}a^{18}+\frac{31}{3179}a^{17}-\frac{3}{187}a^{16}+\frac{32}{3179}a^{15}-\frac{54}{3179}a^{14}+\frac{5}{187}a^{13}-\frac{57}{3179}a^{12}+\frac{84}{3179}a^{11}+\frac{392}{3179}a^{10}+\frac{1412}{3179}a^{9}-\frac{808}{3179}a^{8}-\frac{666}{3179}a^{7}-\frac{8}{17}a^{6}-\frac{892}{3179}a^{5}-\frac{105}{3179}a^{4}-\frac{966}{3179}a^{3}+\frac{310}{3179}a^{2}-\frac{1500}{3179}a-\frac{580}{3179}$, $\frac{1}{3179}a^{23}-\frac{1}{3179}a^{21}-\frac{18}{3179}a^{20}-\frac{26}{3179}a^{19}+\frac{7}{3179}a^{18}+\frac{79}{3179}a^{17}-\frac{2}{3179}a^{16}-\frac{58}{3179}a^{15}-\frac{23}{3179}a^{14}+\frac{28}{3179}a^{13}+\frac{72}{3179}a^{12}+\frac{16}{3179}a^{11}+\frac{921}{3179}a^{10}+\frac{996}{3179}a^{9}-\frac{157}{3179}a^{8}+\frac{113}{3179}a^{7}+\frac{1471}{3179}a^{6}-\frac{852}{3179}a^{5}+\frac{439}{3179}a^{4}+\frac{1234}{3179}a^{3}+\frac{1211}{3179}a^{2}-\frac{61}{3179}a+\frac{727}{3179}$, $\frac{1}{3179}a^{24}-\frac{3}{3179}a^{21}-\frac{91}{3179}a^{20}-\frac{4}{187}a^{19}-\frac{18}{3179}a^{18}-\frac{2}{289}a^{17}+\frac{27}{3179}a^{16}+\frac{60}{3179}a^{15}-\frac{26}{3179}a^{14}-\frac{13}{3179}a^{13}-\frac{24}{3179}a^{12}-\frac{83}{3179}a^{11}-\frac{1536}{3179}a^{10}+\frac{1272}{3179}a^{9}+\frac{2}{3179}a^{8}-\frac{606}{3179}a^{7}-\frac{1549}{3179}a^{6}+\frac{1434}{3179}a^{5}+\frac{619}{3179}a^{4}-\frac{214}{3179}a^{3}-\frac{1366}{3179}a^{2}-\frac{467}{3179}a-\frac{1107}{3179}$, $\frac{1}{3179}a^{25}+\frac{5}{3179}a^{21}+\frac{26}{3179}a^{20}-\frac{2}{289}a^{19}-\frac{92}{3179}a^{18}+\frac{1}{3179}a^{17}-\frac{25}{3179}a^{16}+\frac{2}{3179}a^{15}+\frac{12}{3179}a^{14}-\frac{41}{3179}a^{13}+\frac{35}{3179}a^{12}+\frac{42}{3179}a^{11}+\frac{3}{187}a^{10}-\frac{335}{3179}a^{9}+\frac{404}{3179}a^{8}+\frac{329}{3179}a^{7}-\frac{504}{3179}a^{6}-\frac{49}{187}a^{5}+\frac{1273}{3179}a^{4}-\frac{111}{289}a^{3}+\frac{871}{3179}a^{2}-\frac{405}{3179}a-\frac{1349}{3179}$, $\frac{1}{3179}a^{26}+\frac{2}{3179}a^{21}+\frac{31}{3179}a^{20}-\frac{57}{3179}a^{19}-\frac{41}{3179}a^{18}-\frac{78}{3179}a^{17}-\frac{15}{3179}a^{16}-\frac{63}{3179}a^{15}+\frac{42}{3179}a^{14}-\frac{50}{3179}a^{13}-\frac{81}{3179}a^{12}-\frac{63}{3179}a^{11}-\frac{2}{17}a^{10}+\frac{229}{3179}a^{9}-\frac{67}{187}a^{8}-\frac{336}{3179}a^{7}+\frac{6}{17}a^{6}+\frac{650}{3179}a^{5}+\frac{1072}{3179}a^{4}-\frac{1422}{3179}a^{3}+\frac{9}{187}a^{2}+\frac{116}{3179}a+\frac{1336}{3179}$, $\frac{1}{2323849}a^{27}+\frac{256}{2323849}a^{26}+\frac{15}{211259}a^{25}-\frac{12}{211259}a^{24}+\frac{23}{211259}a^{23}+\frac{2}{2323849}a^{22}+\frac{2174}{2323849}a^{21}-\frac{57213}{2323849}a^{20}+\frac{51897}{2323849}a^{19}-\frac{783}{136697}a^{18}+\frac{51574}{2323849}a^{17}-\frac{17266}{2323849}a^{16}-\frac{63476}{2323849}a^{15}-\frac{60952}{2323849}a^{14}-\frac{214}{136697}a^{13}-\frac{14482}{2323849}a^{12}+\frac{35743}{2323849}a^{11}-\frac{316460}{2323849}a^{10}-\frac{341086}{2323849}a^{9}-\frac{80904}{211259}a^{8}-\frac{781905}{2323849}a^{7}+\frac{1113771}{2323849}a^{6}+\frac{104553}{211259}a^{5}+\frac{8930}{136697}a^{4}-\frac{504408}{2323849}a^{3}+\frac{929658}{2323849}a^{2}-\frac{90703}{211259}a-\frac{15322}{54043}$, $\frac{1}{8744643787}a^{28}-\frac{1385}{8744643787}a^{27}-\frac{48476}{794967617}a^{26}+\frac{319751}{8744643787}a^{25}-\frac{1272913}{8744643787}a^{24}-\frac{953918}{8744643787}a^{23}+\frac{1191884}{8744643787}a^{22}-\frac{23254290}{8744643787}a^{21}-\frac{10412375}{514390811}a^{20}-\frac{252428357}{8744643787}a^{19}+\frac{103882423}{8744643787}a^{18}+\frac{222523310}{8744643787}a^{17}+\frac{47425154}{8744643787}a^{16}+\frac{84369088}{8744643787}a^{15}+\frac{233676827}{8744643787}a^{14}+\frac{133316875}{8744643787}a^{13}+\frac{208612856}{8744643787}a^{12}+\frac{136757451}{8744643787}a^{11}+\frac{1724727227}{8744643787}a^{10}+\frac{72698176}{8744643787}a^{9}+\frac{2401216568}{8744643787}a^{8}+\frac{666368442}{8744643787}a^{7}+\frac{562403001}{8744643787}a^{6}-\frac{2639366288}{8744643787}a^{5}+\frac{1977451678}{8744643787}a^{4}-\frac{580022060}{8744643787}a^{3}+\frac{1786372273}{8744643787}a^{2}-\frac{4247656330}{8744643787}a-\frac{2504990}{203363809}$, $\frac{1}{10869592227241}a^{29}+\frac{394}{10869592227241}a^{28}-\frac{927501}{10869592227241}a^{27}-\frac{294692808}{10869592227241}a^{26}-\frac{17385894}{988144747931}a^{25}+\frac{1040069708}{10869592227241}a^{24}-\frac{919137512}{10869592227241}a^{23}-\frac{1180401683}{10869592227241}a^{22}-\frac{7492882953}{10869592227241}a^{21}+\frac{111183350176}{10869592227241}a^{20}-\frac{1430506986}{96191081657}a^{19}-\frac{2376810727}{153092848271}a^{18}-\frac{249737690075}{10869592227241}a^{17}+\frac{115215844176}{10869592227241}a^{16}-\frac{59682595032}{10869592227241}a^{15}+\frac{319394965449}{10869592227241}a^{14}+\frac{290548062479}{10869592227241}a^{13}-\frac{69194759450}{10869592227241}a^{12}+\frac{14786857864}{988144747931}a^{11}+\frac{885140395022}{10869592227241}a^{10}-\frac{451544042711}{10869592227241}a^{9}+\frac{2228595007301}{10869592227241}a^{8}-\frac{3855946038863}{10869592227241}a^{7}+\frac{3177923235990}{10869592227241}a^{6}+\frac{4707563738929}{10869592227241}a^{5}+\frac{82060550320}{252781214587}a^{4}+\frac{4269053233906}{10869592227241}a^{3}+\frac{920063676781}{10869592227241}a^{2}-\frac{3206359537761}{10869592227241}a+\frac{55691219114}{252781214587}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{5}$, which has order $5$ (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:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{82929283375}{10869592227241}a^{29}-\frac{511299728651}{10869592227241}a^{28}+\frac{3181225099007}{10869592227241}a^{27}-\frac{15009427331757}{10869592227241}a^{26}+\frac{5294950296734}{988144747931}a^{25}-\frac{194510886995199}{10869592227241}a^{24}+\frac{569763043018339}{10869592227241}a^{23}-\frac{14\!\cdots\!37}{10869592227241}a^{22}+\frac{34\!\cdots\!85}{10869592227241}a^{21}-\frac{75\!\cdots\!97}{10869592227241}a^{20}+\frac{133288174831586}{96191081657}a^{19}-\frac{28\!\cdots\!32}{10869592227241}a^{18}+\frac{49\!\cdots\!07}{10869592227241}a^{17}-\frac{82\!\cdots\!07}{10869592227241}a^{16}+\frac{13\!\cdots\!71}{10869592227241}a^{15}-\frac{19\!\cdots\!76}{10869592227241}a^{14}+\frac{28\!\cdots\!82}{10869592227241}a^{13}-\frac{37\!\cdots\!42}{10869592227241}a^{12}+\frac{42\!\cdots\!10}{988144747931}a^{11}-\frac{10\!\cdots\!48}{205086645797}a^{10}+\frac{56\!\cdots\!92}{10869592227241}a^{9}-\frac{53\!\cdots\!05}{10869592227241}a^{8}+\frac{26\!\cdots\!70}{639387778073}a^{7}-\frac{33\!\cdots\!97}{10869592227241}a^{6}+\frac{21\!\cdots\!27}{10869592227241}a^{5}-\frac{11\!\cdots\!26}{10869592227241}a^{4}+\frac{50\!\cdots\!86}{10869592227241}a^{3}-\frac{16\!\cdots\!40}{10869592227241}a^{2}+\frac{34\!\cdots\!49}{10869592227241}a-\frac{8588741322094}{252781214587}$, $\frac{20328240874}{10869592227241}a^{29}-\frac{148472513357}{10869592227241}a^{28}+\frac{52959321538}{639387778073}a^{27}-\frac{4457508829531}{10869592227241}a^{26}+\frac{1613626623883}{988144747931}a^{25}-\frac{60800839681962}{10869592227241}a^{24}+\frac{182225258546029}{10869592227241}a^{23}-\frac{484778586249500}{10869592227241}a^{22}+\frac{11\!\cdots\!78}{10869592227241}a^{21}-\frac{25\!\cdots\!87}{10869592227241}a^{20}+\frac{45916328415095}{96191081657}a^{19}-\frac{98\!\cdots\!04}{10869592227241}a^{18}+\frac{17\!\cdots\!98}{10869592227241}a^{17}-\frac{29\!\cdots\!29}{10869592227241}a^{16}+\frac{47\!\cdots\!15}{10869592227241}a^{15}-\frac{72\!\cdots\!39}{10869592227241}a^{14}+\frac{61\!\cdots\!11}{639387778073}a^{13}-\frac{14\!\cdots\!35}{10869592227241}a^{12}+\frac{16\!\cdots\!93}{988144747931}a^{11}-\frac{21\!\cdots\!74}{10869592227241}a^{10}+\frac{22\!\cdots\!91}{10869592227241}a^{9}-\frac{22\!\cdots\!50}{10869592227241}a^{8}+\frac{19\!\cdots\!81}{10869592227241}a^{7}-\frac{28\!\cdots\!76}{205086645797}a^{6}+\frac{99\!\cdots\!63}{10869592227241}a^{5}-\frac{56\!\cdots\!74}{10869592227241}a^{4}+\frac{25\!\cdots\!71}{10869592227241}a^{3}-\frac{91\!\cdots\!11}{10869592227241}a^{2}+\frac{22\!\cdots\!38}{10869592227241}a-\frac{6586981642693}{252781214587}$, $\frac{64782116428}{10869592227241}a^{29}-\frac{6887026623}{205086645797}a^{28}+\frac{2298430608915}{10869592227241}a^{27}-\frac{10544569499892}{10869592227241}a^{26}+\frac{3650824354020}{988144747931}a^{25}-\frac{131813394777763}{10869592227241}a^{24}+\frac{379685035266609}{10869592227241}a^{23}-\frac{971716127381179}{10869592227241}a^{22}+\frac{22\!\cdots\!11}{10869592227241}a^{21}-\frac{48\!\cdots\!02}{10869592227241}a^{20}+\frac{83899703123792}{96191081657}a^{19}-\frac{17\!\cdots\!21}{10869592227241}a^{18}+\frac{30\!\cdots\!47}{10869592227241}a^{17}-\frac{29\!\cdots\!99}{639387778073}a^{16}+\frac{79\!\cdots\!84}{10869592227241}a^{15}-\frac{11\!\cdots\!20}{10869592227241}a^{14}+\frac{16\!\cdots\!37}{10869592227241}a^{13}-\frac{22\!\cdots\!35}{10869592227241}a^{12}+\frac{464929559835683}{18644240527}a^{11}-\frac{30\!\cdots\!70}{10869592227241}a^{10}+\frac{31\!\cdots\!46}{10869592227241}a^{9}-\frac{17\!\cdots\!62}{639387778073}a^{8}+\frac{23\!\cdots\!66}{10869592227241}a^{7}-\frac{17\!\cdots\!07}{10869592227241}a^{6}+\frac{10\!\cdots\!24}{10869592227241}a^{5}-\frac{53\!\cdots\!96}{10869592227241}a^{4}+\frac{12\!\cdots\!63}{639387778073}a^{3}-\frac{64\!\cdots\!34}{10869592227241}a^{2}+\frac{12\!\cdots\!97}{10869592227241}a-\frac{175531112154}{14869483211}$, $\frac{71188492753}{10869592227241}a^{29}-\frac{344553323166}{10869592227241}a^{28}+\frac{2244893194708}{10869592227241}a^{27}-\frac{9758895029434}{10869592227241}a^{26}+\frac{3280716902091}{988144747931}a^{25}-\frac{114809653841052}{10869592227241}a^{24}+\frac{320607414073633}{10869592227241}a^{23}-\frac{794814784120243}{10869592227241}a^{22}+\frac{17\!\cdots\!05}{10869592227241}a^{21}-\frac{70132975416146}{205086645797}a^{20}+\frac{3715458262779}{5658298921}a^{19}-\frac{12\!\cdots\!67}{10869592227241}a^{18}+\frac{21\!\cdots\!09}{10869592227241}a^{17}-\frac{35\!\cdots\!78}{10869592227241}a^{16}+\frac{54\!\cdots\!60}{10869592227241}a^{15}-\frac{79\!\cdots\!44}{10869592227241}a^{14}+\frac{10\!\cdots\!04}{10869592227241}a^{13}-\frac{13\!\cdots\!06}{10869592227241}a^{12}+\frac{14\!\cdots\!52}{988144747931}a^{11}-\frac{17\!\cdots\!55}{10869592227241}a^{10}+\frac{16\!\cdots\!78}{10869592227241}a^{9}-\frac{14\!\cdots\!70}{10869592227241}a^{8}+\frac{10\!\cdots\!24}{10869592227241}a^{7}-\frac{68\!\cdots\!72}{10869592227241}a^{6}+\frac{20\!\cdots\!46}{639387778073}a^{5}-\frac{13\!\cdots\!31}{10869592227241}a^{4}+\frac{36\!\cdots\!77}{10869592227241}a^{3}-\frac{297374616801179}{10869592227241}a^{2}-\frac{92687734546838}{10869592227241}a+\frac{499434695299}{252781214587}$, $\frac{6477499551}{10869592227241}a^{29}-\frac{13941144911}{10869592227241}a^{28}+\frac{120367797708}{10869592227241}a^{27}-\frac{356518872354}{10869592227241}a^{26}+\frac{89384381715}{988144747931}a^{25}-\frac{2131830783601}{10869592227241}a^{24}+\frac{3250777329945}{10869592227241}a^{23}-\frac{1419691859767}{10869592227241}a^{22}-\frac{492056756890}{639387778073}a^{21}+\frac{2439827033751}{639387778073}a^{20}-\frac{1068709618040}{96191081657}a^{19}+\frac{278051769570480}{10869592227241}a^{18}-\frac{576447784507878}{10869592227241}a^{17}+\frac{10\!\cdots\!05}{10869592227241}a^{16}-\frac{19\!\cdots\!46}{10869592227241}a^{15}+\frac{190956046710285}{639387778073}a^{14}-\frac{51\!\cdots\!86}{10869592227241}a^{13}+\frac{75\!\cdots\!34}{10869592227241}a^{12}-\frac{940569972908507}{988144747931}a^{11}+\frac{12\!\cdots\!60}{10869592227241}a^{10}-\frac{14\!\cdots\!52}{10869592227241}a^{9}+\frac{15\!\cdots\!31}{10869592227241}a^{8}-\frac{49311560261316}{37611045769}a^{7}+\frac{11\!\cdots\!96}{10869592227241}a^{6}-\frac{209917301770746}{252781214587}a^{5}+\frac{60\!\cdots\!44}{10869592227241}a^{4}-\frac{35\!\cdots\!66}{10869592227241}a^{3}+\frac{16\!\cdots\!14}{10869592227241}a^{2}-\frac{536603559902952}{10869592227241}a+\frac{1853964715819}{252781214587}$, $\frac{328378203}{988144747931}a^{29}-\frac{8996309}{5284196513}a^{28}+\frac{10637303747}{988144747931}a^{27}-\frac{47100798314}{988144747931}a^{26}+\frac{174289250486}{988144747931}a^{25}-\frac{1922021212}{3419185979}a^{24}+\frac{1556493797241}{988144747931}a^{23}-\frac{352170098938}{89831340721}a^{22}+\frac{8792323907784}{988144747931}a^{21}-\frac{18336475099651}{988144747931}a^{20}+\frac{18499938350}{514390811}a^{19}-\frac{64854133670925}{988144747931}a^{18}+\frac{6591213303068}{58126161643}a^{17}-\frac{183918836463789}{988144747931}a^{16}+\frac{287545536073402}{988144747931}a^{15}-\frac{426226091754376}{988144747931}a^{14}+\frac{596071800790834}{988144747931}a^{13}-\frac{18194590572683}{22980110417}a^{12}+\frac{956967982584292}{988144747931}a^{11}-\frac{10\!\cdots\!62}{988144747931}a^{10}+\frac{103245436536439}{89831340721}a^{9}-\frac{64228262020225}{58126161643}a^{8}+\frac{962011472013534}{988144747931}a^{7}-\frac{70234932302481}{89831340721}a^{6}+\frac{33003576666076}{58126161643}a^{5}-\frac{361373043026966}{988144747931}a^{4}+\frac{201980803521737}{988144747931}a^{3}-\frac{156999243756}{1694930957}a^{2}+\frac{30254583678592}{988144747931}a-\frac{118782296514}{22980110417}$, $\frac{26517394696}{10869592227241}a^{29}-\frac{177828748682}{10869592227241}a^{28}+\frac{1100177629306}{10869592227241}a^{27}-\frac{5326492532837}{10869592227241}a^{26}+\frac{1913484610797}{988144747931}a^{25}-\frac{71541478383896}{10869592227241}a^{24}+\frac{213000304015679}{10869592227241}a^{23}-\frac{563160165450255}{10869592227241}a^{22}+\frac{13\!\cdots\!49}{10869592227241}a^{21}-\frac{29\!\cdots\!45}{10869592227241}a^{20}+\frac{52654665746148}{96191081657}a^{19}-\frac{11\!\cdots\!81}{10869592227241}a^{18}+\frac{19\!\cdots\!58}{10869592227241}a^{17}-\frac{33\!\cdots\!43}{10869592227241}a^{16}+\frac{53\!\cdots\!73}{10869592227241}a^{15}-\frac{82\!\cdots\!48}{10869592227241}a^{14}+\frac{11\!\cdots\!14}{10869592227241}a^{13}-\frac{16\!\cdots\!42}{10869592227241}a^{12}+\frac{10\!\cdots\!30}{58126161643}a^{11}-\frac{23\!\cdots\!38}{10869592227241}a^{10}+\frac{25\!\cdots\!31}{10869592227241}a^{9}-\frac{24\!\cdots\!61}{10869592227241}a^{8}+\frac{21\!\cdots\!13}{10869592227241}a^{7}-\frac{16\!\cdots\!68}{10869592227241}a^{6}+\frac{10\!\cdots\!13}{10869592227241}a^{5}-\frac{61\!\cdots\!88}{10869592227241}a^{4}+\frac{28\!\cdots\!72}{10869592227241}a^{3}-\frac{99\!\cdots\!54}{10869592227241}a^{2}+\frac{23\!\cdots\!77}{10869592227241}a-\frac{6453735712223}{252781214587}$, $\frac{85612187037}{10869592227241}a^{29}-\frac{6690473503}{153092848271}a^{28}+\frac{3028481059232}{10869592227241}a^{27}-\frac{13810994871727}{10869592227241}a^{26}+\frac{4801064064334}{988144747931}a^{25}-\frac{173524148321503}{10869592227241}a^{24}+\frac{500943780987948}{10869592227241}a^{23}-\frac{12\!\cdots\!78}{10869592227241}a^{22}+\frac{29\!\cdots\!31}{10869592227241}a^{21}-\frac{376226632606645}{639387778073}a^{20}+\frac{111955316577875}{96191081657}a^{19}-\frac{442477743858756}{205086645797}a^{18}+\frac{41\!\cdots\!13}{10869592227241}a^{17}-\frac{12\!\cdots\!98}{205086645797}a^{16}+\frac{10\!\cdots\!42}{10869592227241}a^{15}-\frac{37\!\cdots\!06}{252781214587}a^{14}+\frac{22\!\cdots\!70}{10869592227241}a^{13}-\frac{30\!\cdots\!26}{10869592227241}a^{12}+\frac{33\!\cdots\!06}{988144747931}a^{11}-\frac{42\!\cdots\!33}{10869592227241}a^{10}+\frac{43\!\cdots\!24}{10869592227241}a^{9}-\frac{40\!\cdots\!96}{10869592227241}a^{8}+\frac{34\!\cdots\!82}{10869592227241}a^{7}-\frac{24\!\cdots\!61}{10869592227241}a^{6}+\frac{15\!\cdots\!82}{10869592227241}a^{5}-\frac{81\!\cdots\!71}{10869592227241}a^{4}+\frac{34\!\cdots\!71}{10869592227241}a^{3}-\frac{10\!\cdots\!77}{10869592227241}a^{2}+\frac{20\!\cdots\!75}{10869592227241}a-\frac{4045477962165}{252781214587}$, $\frac{2458235570}{988144747931}a^{29}-\frac{13023920725}{988144747931}a^{28}+\frac{7578330785}{89831340721}a^{27}-\frac{374772219776}{988144747931}a^{26}+\frac{128650798973}{89831340721}a^{25}-\frac{4602455193020}{988144747931}a^{24}+\frac{13147664651320}{988144747931}a^{23}-\frac{33378237316372}{988144747931}a^{22}+\frac{4534688570116}{58126161643}a^{21}-\frac{163171710722829}{988144747931}a^{20}+\frac{2833119645468}{8744643787}a^{19}-\frac{589424118197433}{988144747931}a^{18}+\frac{10\!\cdots\!34}{988144747931}a^{17}-\frac{16\!\cdots\!79}{988144747931}a^{16}+\frac{26\!\cdots\!97}{988144747931}a^{15}-\frac{91707575141516}{22980110417}a^{14}+\frac{55\!\cdots\!74}{988144747931}a^{13}-\frac{72\!\cdots\!42}{988144747931}a^{12}+\frac{88\!\cdots\!10}{988144747931}a^{11}-\frac{99\!\cdots\!91}{988144747931}a^{10}+\frac{10\!\cdots\!43}{988144747931}a^{9}-\frac{93\!\cdots\!55}{988144747931}a^{8}+\frac{76\!\cdots\!80}{988144747931}a^{7}-\frac{55\!\cdots\!40}{988144747931}a^{6}+\frac{33\!\cdots\!38}{988144747931}a^{5}-\frac{17\!\cdots\!90}{988144747931}a^{4}+\frac{62774655126245}{89831340721}a^{3}-\frac{198991443966133}{988144747931}a^{2}+\frac{35424951250616}{988144747931}a-\frac{3055445784}{1351771201}$, $\frac{100971515451}{10869592227241}a^{29}-\frac{32125628166}{639387778073}a^{28}+\frac{3502552549407}{10869592227241}a^{27}-\frac{15838065840829}{10869592227241}a^{26}+\frac{5481010013949}{988144747931}a^{25}-\frac{197216202485428}{10869592227241}a^{24}+\frac{566708007187327}{10869592227241}a^{23}-\frac{14\!\cdots\!27}{10869592227241}a^{22}+\frac{33\!\cdots\!42}{10869592227241}a^{21}-\frac{71\!\cdots\!69}{10869592227241}a^{20}+\frac{124521470460249}{96191081657}a^{19}-\frac{25\!\cdots\!29}{10869592227241}a^{18}+\frac{45\!\cdots\!18}{10869592227241}a^{17}-\frac{74\!\cdots\!18}{10869592227241}a^{16}+\frac{27\!\cdots\!58}{252781214587}a^{15}-\frac{17\!\cdots\!77}{10869592227241}a^{14}+\frac{24\!\cdots\!96}{10869592227241}a^{13}-\frac{32\!\cdots\!35}{10869592227241}a^{12}+\frac{36\!\cdots\!67}{988144747931}a^{11}-\frac{45\!\cdots\!20}{10869592227241}a^{10}+\frac{46\!\cdots\!41}{10869592227241}a^{9}-\frac{43\!\cdots\!59}{10869592227241}a^{8}+\frac{35\!\cdots\!26}{10869592227241}a^{7}-\frac{25\!\cdots\!42}{10869592227241}a^{6}+\frac{22\!\cdots\!28}{153092848271}a^{5}-\frac{82\!\cdots\!19}{10869592227241}a^{4}+\frac{33\!\cdots\!57}{10869592227241}a^{3}-\frac{99\!\cdots\!93}{10869592227241}a^{2}+\frac{17\!\cdots\!96}{10869592227241}a-\frac{2611860606926}{252781214587}$, $\frac{182807441596}{10869592227241}a^{29}-\frac{982980305294}{10869592227241}a^{28}+\frac{6320347734539}{10869592227241}a^{27}-\frac{28546035198491}{10869592227241}a^{26}+\frac{9881944810680}{988144747931}a^{25}-\frac{355736291576233}{10869592227241}a^{24}+\frac{10\!\cdots\!56}{10869592227241}a^{23}-\frac{49351810994337}{205086645797}a^{22}+\frac{60\!\cdots\!98}{10869592227241}a^{21}-\frac{12\!\cdots\!83}{10869592227241}a^{20}+\frac{225945068879988}{96191081657}a^{19}-\frac{47\!\cdots\!94}{10869592227241}a^{18}+\frac{82\!\cdots\!07}{10869592227241}a^{17}-\frac{13\!\cdots\!94}{10869592227241}a^{16}+\frac{21\!\cdots\!12}{10869592227241}a^{15}-\frac{32\!\cdots\!67}{10869592227241}a^{14}+\frac{45\!\cdots\!78}{10869592227241}a^{13}-\frac{59\!\cdots\!94}{10869592227241}a^{12}+\frac{66\!\cdots\!77}{988144747931}a^{11}-\frac{83\!\cdots\!54}{10869592227241}a^{10}+\frac{85\!\cdots\!57}{10869592227241}a^{9}-\frac{79\!\cdots\!51}{10869592227241}a^{8}+\frac{65\!\cdots\!35}{10869592227241}a^{7}-\frac{47\!\cdots\!97}{10869592227241}a^{6}+\frac{29\!\cdots\!41}{10869592227241}a^{5}-\frac{15\!\cdots\!42}{10869592227241}a^{4}+\frac{11\!\cdots\!09}{205086645797}a^{3}-\frac{19\!\cdots\!06}{10869592227241}a^{2}+\frac{36\!\cdots\!59}{10869592227241}a-\frac{6494053750902}{252781214587}$, $\frac{78259850}{5284196513}a^{29}-\frac{86579954308}{988144747931}a^{28}+\frac{547478479880}{988144747931}a^{27}-\frac{232088997682}{89831340721}a^{26}+\frac{582344336014}{58126161643}a^{25}-\frac{33016932420247}{988144747931}a^{24}+\frac{96658648011130}{988144747931}a^{23}-\frac{251548335765054}{988144747931}a^{22}+\frac{593704575750965}{988144747931}a^{21}-\frac{12\!\cdots\!95}{988144747931}a^{20}+\frac{22717425901149}{8744643787}a^{19}-\frac{67699649025830}{13917531661}a^{18}+\frac{17939997734869}{2089100947}a^{17}-\frac{835340848041131}{58126161643}a^{16}+\frac{22\!\cdots\!92}{988144747931}a^{15}-\frac{34\!\cdots\!07}{988144747931}a^{14}+\frac{48\!\cdots\!28}{988144747931}a^{13}-\frac{59\!\cdots\!61}{89831340721}a^{12}+\frac{82\!\cdots\!88}{988144747931}a^{11}-\frac{22\!\cdots\!96}{22980110417}a^{10}+\frac{91\!\cdots\!15}{89831340721}a^{9}-\frac{95\!\cdots\!76}{988144747931}a^{8}+\frac{82\!\cdots\!95}{988144747931}a^{7}-\frac{61\!\cdots\!70}{988144747931}a^{6}+\frac{40\!\cdots\!39}{988144747931}a^{5}-\frac{22\!\cdots\!76}{988144747931}a^{4}+\frac{99\!\cdots\!61}{988144747931}a^{3}-\frac{33\!\cdots\!11}{988144747931}a^{2}+\frac{760715845853895}{988144747931}a-\frac{181431806136}{2089100947}$, $\frac{21523043877}{10869592227241}a^{29}-\frac{88896862108}{10869592227241}a^{28}+\frac{600651647185}{10869592227241}a^{27}-\frac{2439634662843}{10869592227241}a^{26}+\frac{785511895489}{988144747931}a^{25}-\frac{26106410599080}{10869592227241}a^{24}+\frac{68915885355557}{10869592227241}a^{23}-\frac{160094112076047}{10869592227241}a^{22}+\frac{4772622410891}{153092848271}a^{21}-\frac{650716811699629}{10869592227241}a^{20}+\frac{10214757328708}{96191081657}a^{19}-\frac{19\!\cdots\!08}{10869592227241}a^{18}+\frac{30\!\cdots\!23}{10869592227241}a^{17}-\frac{44\!\cdots\!95}{10869592227241}a^{16}+\frac{61\!\cdots\!68}{10869592227241}a^{15}-\frac{76\!\cdots\!54}{10869592227241}a^{14}+\frac{85\!\cdots\!66}{10869592227241}a^{13}-\frac{454244525678116}{639387778073}a^{12}+\frac{378656962014466}{988144747931}a^{11}+\frac{24\!\cdots\!70}{10869592227241}a^{10}-\frac{654779481639632}{639387778073}a^{9}+\frac{20\!\cdots\!10}{10869592227241}a^{8}-\frac{25\!\cdots\!38}{10869592227241}a^{7}+\frac{27\!\cdots\!08}{10869592227241}a^{6}-\frac{439437434330566}{205086645797}a^{5}+\frac{16\!\cdots\!89}{10869592227241}a^{4}-\frac{91\!\cdots\!39}{10869592227241}a^{3}+\frac{39\!\cdots\!09}{10869592227241}a^{2}-\frac{11\!\cdots\!04}{10869592227241}a+\frac{3322383372730}{252781214587}$, $\frac{19293805268}{10869592227241}a^{29}-\frac{1738896111}{153092848271}a^{28}+\frac{753294799650}{10869592227241}a^{27}-\frac{3586505206837}{10869592227241}a^{26}+\frac{1262179806896}{988144747931}a^{25}-\frac{46258030299188}{10869592227241}a^{24}+\frac{135149778985863}{10869592227241}a^{23}-\frac{350540954645793}{10869592227241}a^{22}+\frac{822829231405318}{10869592227241}a^{21}-\frac{17\!\cdots\!28}{10869592227241}a^{20}+\frac{31146631421767}{96191081657}a^{19}-\frac{65\!\cdots\!11}{10869592227241}a^{18}+\frac{11\!\cdots\!25}{10869592227241}a^{17}-\frac{19\!\cdots\!09}{10869592227241}a^{16}+\frac{30\!\cdots\!31}{10869592227241}a^{15}-\frac{45\!\cdots\!23}{10869592227241}a^{14}+\frac{64\!\cdots\!13}{10869592227241}a^{13}-\frac{86\!\cdots\!63}{10869592227241}a^{12}+\frac{96\!\cdots\!93}{988144747931}a^{11}-\frac{12\!\cdots\!34}{10869592227241}a^{10}+\frac{12\!\cdots\!69}{10869592227241}a^{9}-\frac{11\!\cdots\!18}{10869592227241}a^{8}+\frac{99\!\cdots\!19}{10869592227241}a^{7}-\frac{73\!\cdots\!74}{10869592227241}a^{6}+\frac{46\!\cdots\!69}{10869592227241}a^{5}-\frac{25\!\cdots\!61}{10869592227241}a^{4}+\frac{11\!\cdots\!38}{10869592227241}a^{3}-\frac{37\!\cdots\!46}{10869592227241}a^{2}+\frac{822359763200090}{10869592227241}a-\frac{2231127608586}{252781214587}$ Copy content Toggle raw display (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:  \( 588337897727.5111 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 588337897727.5111 \cdot 5}{2\cdot\sqrt{6072125144349637560639895035214067242224098130847}}\cr\approx \mathstrut & 0.560524373000263 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849)
 
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^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849, 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^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849);
 
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^30 - 6*x^29 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849);
 
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

$D_{30}$ (as 30T14):

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 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-127}) \), 3.1.2159.1, 5.1.4661281.1, 6.0.591982687.1, 10.0.2759397651242047.1, 15.1.218659573334046061397519.1

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 30 sibling: 30.2.812804153180660145912426894477473567856769041137.1
Minimal sibling: 30.2.812804153180660145912426894477473567856769041137.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ $30$ ${\href{/padicField/5.6.0.1}{6} }^{5}$ ${\href{/padicField/7.10.0.1}{10} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{14}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{6}$ R $15^{2}$ $30$ $30$ ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{14}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{15}$ ${\href{/padicField/47.5.0.1}{5} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{15}$ ${\href{/padicField/59.2.0.1}{2} }^{15}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
\(127\) Copy content Toggle raw display 127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 127$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.17.2t1.a.a$1$ $ 17 $ \(\Q(\sqrt{17}) \) $C_2$ (as 2T1) $1$ $1$
1.2159.2t1.a.a$1$ $ 17 \cdot 127 $ \(\Q(\sqrt{-2159}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.127.2t1.a.a$1$ $ 127 $ \(\Q(\sqrt{-127}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2159.6t3.b.a$2$ $ 17 \cdot 127 $ 6.2.79241777.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.2159.3t2.a.a$2$ $ 17 \cdot 127 $ 3.1.2159.1 $S_3$ (as 3T2) $1$ $0$
* 2.2159.5t2.a.a$2$ $ 17 \cdot 127 $ 5.1.4661281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2159.5t2.a.b$2$ $ 17 \cdot 127 $ 5.1.4661281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2159.10t3.a.a$2$ $ 17 \cdot 127 $ 10.2.369368189536337.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.2159.10t3.a.b$2$ $ 17 \cdot 127 $ 10.2.369368189536337.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.2159.15t2.a.c$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.d$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.a$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.b$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.d$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.a$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.b$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.a.c$2$ $ 17 \cdot 127 $ 30.0.6072125144349637560639895035214067242224098130847.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.