Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1)
gp: K = bnfinit(y^24 - 8*y^23 + 35*y^22 - 105*y^21 + 233*y^20 - 377*y^19 + 373*y^18 + 86*y^17 - 1310*y^16 + 3226*y^15 - 4893*y^14 + 4424*y^13 - 205*y^12 - 6944*y^11 + 13153*y^10 - 14094*y^9 + 9116*y^8 - 2660*y^7 - 525*y^6 + 689*y^5 - 175*y^4 - 37*y^3 + 37*y^2 - 10*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1)
\( x^{24} - 8 x^{23} + 35 x^{22} - 105 x^{21} + 233 x^{20} - 377 x^{19} + 373 x^{18} + 86 x^{17} - 1310 x^{16} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1729054511370401309967041015625\)
\(\medspace = 3^{12}\cdot 5^{18}\cdot 31^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.19\) | 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: | $3^{1/2}5^{3/4}31^{2/3}\approx 57.15171403472212$ | ||
| Ramified primes: |
\(3\), \(5\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $12$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{29}a^{20}-\frac{6}{29}a^{19}+\frac{14}{29}a^{18}-\frac{14}{29}a^{17}-\frac{3}{29}a^{16}+\frac{8}{29}a^{15}+\frac{14}{29}a^{14}+\frac{1}{29}a^{13}-\frac{2}{29}a^{12}+\frac{12}{29}a^{11}-\frac{14}{29}a^{10}+\frac{8}{29}a^{9}+\frac{14}{29}a^{8}+\frac{3}{29}a^{7}+\frac{12}{29}a^{6}-\frac{14}{29}a^{5}+\frac{2}{29}a^{4}-\frac{12}{29}a^{3}-\frac{14}{29}a^{2}-\frac{10}{29}a-\frac{1}{29}$, $\frac{1}{899}a^{21}+\frac{5}{899}a^{20}+\frac{209}{899}a^{19}+\frac{314}{899}a^{18}-\frac{418}{899}a^{17}+\frac{381}{899}a^{16}+\frac{131}{899}a^{15}+\frac{5}{29}a^{14}+\frac{154}{899}a^{13}-\frac{10}{899}a^{12}-\frac{172}{899}a^{11}+\frac{376}{899}a^{10}-\frac{391}{899}a^{9}-\frac{336}{899}a^{8}+\frac{364}{899}a^{7}+\frac{2}{899}a^{6}-\frac{210}{899}a^{5}-\frac{222}{899}a^{4}-\frac{88}{899}a^{3}-\frac{193}{899}a^{2}-\frac{169}{899}a-\frac{69}{899}$, $\frac{1}{899}a^{22}-\frac{2}{899}a^{20}+\frac{385}{899}a^{19}-\frac{97}{899}a^{18}-\frac{11}{31}a^{17}-\frac{317}{899}a^{16}-\frac{190}{899}a^{15}+\frac{371}{899}a^{14}-\frac{67}{899}a^{13}+\frac{250}{899}a^{12}-\frac{97}{899}a^{11}+\frac{333}{899}a^{10}+\frac{131}{899}a^{9}+\frac{339}{899}a^{8}+\frac{321}{899}a^{7}+\frac{245}{899}a^{6}-\frac{164}{899}a^{5}-\frac{249}{899}a^{4}-\frac{218}{899}a^{3}-\frac{196}{899}a^{2}-\frac{61}{899}a-\frac{368}{899}$, $\frac{1}{11\!\cdots\!21}a^{23}+\frac{46\!\cdots\!73}{11\!\cdots\!21}a^{22}-\frac{22\!\cdots\!24}{11\!\cdots\!21}a^{21}-\frac{14\!\cdots\!15}{40\!\cdots\!49}a^{20}+\frac{15\!\cdots\!35}{40\!\cdots\!49}a^{19}+\frac{24\!\cdots\!62}{11\!\cdots\!21}a^{18}-\frac{30\!\cdots\!18}{11\!\cdots\!21}a^{17}+\frac{24\!\cdots\!58}{11\!\cdots\!21}a^{16}+\frac{58\!\cdots\!34}{11\!\cdots\!21}a^{15}-\frac{51\!\cdots\!34}{11\!\cdots\!21}a^{14}-\frac{48\!\cdots\!93}{11\!\cdots\!21}a^{13}+\frac{79\!\cdots\!95}{11\!\cdots\!21}a^{12}-\frac{23\!\cdots\!93}{11\!\cdots\!21}a^{11}+\frac{47\!\cdots\!00}{11\!\cdots\!21}a^{10}+\frac{10\!\cdots\!87}{11\!\cdots\!21}a^{9}-\frac{12\!\cdots\!79}{38\!\cdots\!91}a^{8}+\frac{56\!\cdots\!42}{11\!\cdots\!21}a^{7}-\frac{57\!\cdots\!43}{11\!\cdots\!21}a^{6}+\frac{57\!\cdots\!32}{11\!\cdots\!21}a^{5}-\frac{24\!\cdots\!58}{11\!\cdots\!21}a^{4}+\frac{34\!\cdots\!76}{11\!\cdots\!21}a^{3}+\frac{49\!\cdots\!13}{11\!\cdots\!21}a^{2}+\frac{51\!\cdots\!71}{11\!\cdots\!21}a-\frac{42\!\cdots\!72}{11\!\cdots\!21}$
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
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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{7734100806656053550843}{409260124418742790049} a^{23} + \frac{1693683224712430097540188}{11868543608143540911421} a^{22} - \frac{7090008489812080887569001}{11868543608143540911421} a^{21} + \frac{20368041693402810603857679}{11868543608143540911421} a^{20} - \frac{43116509401361366582348538}{11868543608143540911421} a^{19} + \frac{65201825978955000247187270}{11868543608143540911421} a^{18} - \frac{54389259790109903136846857}{11868543608143540911421} a^{17} - \frac{43706410432866229062553548}{11868543608143540911421} a^{16} + \frac{274199302232129549703642975}{11868543608143540911421} a^{15} - \frac{600461831108203963506430442}{11868543608143540911421} a^{14} + \frac{827882160417212094168984537}{11868543608143540911421} a^{13} - \frac{620600631723841736157137884}{11868543608143540911421} a^{12} - \frac{232597302108791818065380974}{11868543608143540911421} a^{11} + \frac{1452958002905821036039530273}{11868543608143540911421} a^{10} - \frac{2297679537017594890047330281}{11868543608143540911421} a^{9} + \frac{2368871237152426530958445}{13201939497378799679} a^{8} - \frac{1088778805324366041663867783}{11868543608143540911421} a^{7} + \frac{3731875305378225621736111}{409260124418742790049} a^{6} + \frac{165980947158962287342034348}{11868543608143540911421} a^{5} - \frac{80030226322022007352319547}{11868543608143540911421} a^{4} + \frac{3587790200741572434146851}{11868543608143540911421} a^{3} + \frac{9838687081568899957579250}{11868543608143540911421} a^{2} - \frac{3879677771549136926752282}{11868543608143540911421} a + \frac{499762829749894184209711}{11868543608143540911421} \)
(order $30$)
| 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: |
$\frac{46\!\cdots\!58}{11\!\cdots\!21}a^{23}-\frac{35\!\cdots\!24}{11\!\cdots\!21}a^{22}+\frac{14\!\cdots\!63}{11\!\cdots\!21}a^{21}-\frac{43\!\cdots\!37}{11\!\cdots\!21}a^{20}+\frac{91\!\cdots\!74}{11\!\cdots\!21}a^{19}-\frac{14\!\cdots\!32}{11\!\cdots\!21}a^{18}+\frac{12\!\cdots\!81}{11\!\cdots\!21}a^{17}+\frac{84\!\cdots\!83}{11\!\cdots\!21}a^{16}-\frac{57\!\cdots\!98}{11\!\cdots\!21}a^{15}+\frac{12\!\cdots\!09}{11\!\cdots\!21}a^{14}-\frac{17\!\cdots\!00}{11\!\cdots\!21}a^{13}+\frac{13\!\cdots\!25}{11\!\cdots\!21}a^{12}+\frac{42\!\cdots\!35}{11\!\cdots\!21}a^{11}-\frac{30\!\cdots\!47}{11\!\cdots\!21}a^{10}+\frac{49\!\cdots\!49}{11\!\cdots\!21}a^{9}-\frac{15\!\cdots\!66}{38\!\cdots\!91}a^{8}+\frac{24\!\cdots\!49}{11\!\cdots\!21}a^{7}-\frac{30\!\cdots\!94}{11\!\cdots\!21}a^{6}-\frac{35\!\cdots\!31}{11\!\cdots\!21}a^{5}+\frac{63\!\cdots\!89}{40\!\cdots\!49}a^{4}-\frac{11\!\cdots\!34}{11\!\cdots\!21}a^{3}-\frac{21\!\cdots\!68}{11\!\cdots\!21}a^{2}+\frac{90\!\cdots\!93}{11\!\cdots\!21}a-\frac{12\!\cdots\!82}{11\!\cdots\!21}$, $\frac{77\!\cdots\!43}{40\!\cdots\!49}a^{23}-\frac{16\!\cdots\!88}{11\!\cdots\!21}a^{22}+\frac{70\!\cdots\!01}{11\!\cdots\!21}a^{21}-\frac{20\!\cdots\!79}{11\!\cdots\!21}a^{20}+\frac{43\!\cdots\!38}{11\!\cdots\!21}a^{19}-\frac{65\!\cdots\!70}{11\!\cdots\!21}a^{18}+\frac{54\!\cdots\!57}{11\!\cdots\!21}a^{17}+\frac{43\!\cdots\!48}{11\!\cdots\!21}a^{16}-\frac{27\!\cdots\!75}{11\!\cdots\!21}a^{15}+\frac{60\!\cdots\!42}{11\!\cdots\!21}a^{14}-\frac{82\!\cdots\!37}{11\!\cdots\!21}a^{13}+\frac{62\!\cdots\!84}{11\!\cdots\!21}a^{12}+\frac{23\!\cdots\!74}{11\!\cdots\!21}a^{11}-\frac{14\!\cdots\!73}{11\!\cdots\!21}a^{10}+\frac{22\!\cdots\!81}{11\!\cdots\!21}a^{9}-\frac{23\!\cdots\!45}{13\!\cdots\!79}a^{8}+\frac{10\!\cdots\!83}{11\!\cdots\!21}a^{7}-\frac{37\!\cdots\!11}{40\!\cdots\!49}a^{6}-\frac{16\!\cdots\!48}{11\!\cdots\!21}a^{5}+\frac{80\!\cdots\!47}{11\!\cdots\!21}a^{4}-\frac{35\!\cdots\!51}{11\!\cdots\!21}a^{3}-\frac{98\!\cdots\!50}{11\!\cdots\!21}a^{2}+\frac{38\!\cdots\!82}{11\!\cdots\!21}a-\frac{51\!\cdots\!32}{11\!\cdots\!21}$, $\frac{13\!\cdots\!58}{11\!\cdots\!21}a^{23}-\frac{10\!\cdots\!96}{11\!\cdots\!21}a^{22}+\frac{43\!\cdots\!66}{11\!\cdots\!21}a^{21}-\frac{12\!\cdots\!28}{11\!\cdots\!21}a^{20}+\frac{27\!\cdots\!14}{11\!\cdots\!21}a^{19}-\frac{43\!\cdots\!40}{11\!\cdots\!21}a^{18}+\frac{39\!\cdots\!42}{11\!\cdots\!21}a^{17}+\frac{19\!\cdots\!98}{11\!\cdots\!21}a^{16}-\frac{16\!\cdots\!00}{11\!\cdots\!21}a^{15}+\frac{38\!\cdots\!32}{11\!\cdots\!21}a^{14}-\frac{56\!\cdots\!54}{11\!\cdots\!21}a^{13}+\frac{46\!\cdots\!16}{11\!\cdots\!21}a^{12}+\frac{75\!\cdots\!82}{11\!\cdots\!21}a^{11}-\frac{90\!\cdots\!44}{11\!\cdots\!21}a^{10}+\frac{15\!\cdots\!60}{11\!\cdots\!21}a^{9}-\frac{49\!\cdots\!04}{38\!\cdots\!91}a^{8}+\frac{86\!\cdots\!91}{11\!\cdots\!21}a^{7}-\frac{14\!\cdots\!26}{11\!\cdots\!21}a^{6}-\frac{11\!\cdots\!26}{11\!\cdots\!21}a^{5}+\frac{67\!\cdots\!46}{11\!\cdots\!21}a^{4}-\frac{67\!\cdots\!52}{11\!\cdots\!21}a^{3}-\frac{70\!\cdots\!78}{11\!\cdots\!21}a^{2}+\frac{33\!\cdots\!70}{11\!\cdots\!21}a-\frac{49\!\cdots\!61}{11\!\cdots\!21}$, $\frac{88\!\cdots\!36}{11\!\cdots\!21}a^{23}-\frac{66\!\cdots\!28}{11\!\cdots\!21}a^{22}+\frac{28\!\cdots\!00}{11\!\cdots\!21}a^{21}-\frac{81\!\cdots\!92}{11\!\cdots\!21}a^{20}+\frac{17\!\cdots\!12}{11\!\cdots\!21}a^{19}-\frac{26\!\cdots\!08}{11\!\cdots\!21}a^{18}+\frac{22\!\cdots\!92}{11\!\cdots\!21}a^{17}+\frac{16\!\cdots\!24}{11\!\cdots\!21}a^{16}-\frac{10\!\cdots\!03}{11\!\cdots\!21}a^{15}+\frac{23\!\cdots\!72}{11\!\cdots\!21}a^{14}-\frac{33\!\cdots\!92}{11\!\cdots\!21}a^{13}+\frac{25\!\cdots\!52}{11\!\cdots\!21}a^{12}+\frac{86\!\cdots\!64}{11\!\cdots\!21}a^{11}-\frac{57\!\cdots\!84}{11\!\cdots\!21}a^{10}+\frac{92\!\cdots\!12}{11\!\cdots\!21}a^{9}-\frac{27\!\cdots\!80}{38\!\cdots\!91}a^{8}+\frac{44\!\cdots\!24}{11\!\cdots\!21}a^{7}-\frac{48\!\cdots\!76}{11\!\cdots\!21}a^{6}-\frac{67\!\cdots\!20}{11\!\cdots\!21}a^{5}+\frac{33\!\cdots\!16}{11\!\cdots\!21}a^{4}-\frac{16\!\cdots\!48}{11\!\cdots\!21}a^{3}-\frac{40\!\cdots\!80}{11\!\cdots\!21}a^{2}+\frac{16\!\cdots\!64}{11\!\cdots\!21}a-\frac{21\!\cdots\!29}{11\!\cdots\!21}$, $\frac{33\!\cdots\!85}{11\!\cdots\!21}a^{23}-\frac{26\!\cdots\!12}{11\!\cdots\!21}a^{22}+\frac{11\!\cdots\!00}{11\!\cdots\!21}a^{21}-\frac{34\!\cdots\!48}{11\!\cdots\!21}a^{20}+\frac{75\!\cdots\!86}{11\!\cdots\!21}a^{19}-\frac{11\!\cdots\!34}{11\!\cdots\!21}a^{18}+\frac{11\!\cdots\!97}{11\!\cdots\!21}a^{17}+\frac{42\!\cdots\!47}{11\!\cdots\!21}a^{16}-\frac{44\!\cdots\!93}{11\!\cdots\!21}a^{15}+\frac{10\!\cdots\!84}{11\!\cdots\!21}a^{14}-\frac{15\!\cdots\!02}{11\!\cdots\!21}a^{13}+\frac{13\!\cdots\!85}{11\!\cdots\!21}a^{12}+\frac{35\!\cdots\!66}{40\!\cdots\!49}a^{11}-\frac{23\!\cdots\!41}{11\!\cdots\!21}a^{10}+\frac{42\!\cdots\!56}{11\!\cdots\!21}a^{9}-\frac{13\!\cdots\!32}{38\!\cdots\!91}a^{8}+\frac{25\!\cdots\!49}{11\!\cdots\!21}a^{7}-\frac{51\!\cdots\!30}{11\!\cdots\!21}a^{6}-\frac{30\!\cdots\!93}{11\!\cdots\!21}a^{5}+\frac{20\!\cdots\!42}{11\!\cdots\!21}a^{4}-\frac{25\!\cdots\!86}{11\!\cdots\!21}a^{3}-\frac{19\!\cdots\!33}{11\!\cdots\!21}a^{2}+\frac{10\!\cdots\!21}{11\!\cdots\!21}a-\frac{16\!\cdots\!47}{11\!\cdots\!21}$, $\frac{16\!\cdots\!19}{11\!\cdots\!21}a^{23}-\frac{12\!\cdots\!82}{11\!\cdots\!21}a^{22}+\frac{52\!\cdots\!43}{11\!\cdots\!21}a^{21}-\frac{14\!\cdots\!18}{11\!\cdots\!21}a^{20}+\frac{31\!\cdots\!20}{11\!\cdots\!21}a^{19}-\frac{47\!\cdots\!75}{11\!\cdots\!21}a^{18}+\frac{38\!\cdots\!64}{11\!\cdots\!21}a^{17}+\frac{33\!\cdots\!47}{11\!\cdots\!21}a^{16}-\frac{20\!\cdots\!50}{11\!\cdots\!21}a^{15}+\frac{43\!\cdots\!27}{11\!\cdots\!21}a^{14}-\frac{59\!\cdots\!08}{11\!\cdots\!21}a^{13}+\frac{44\!\cdots\!91}{11\!\cdots\!21}a^{12}+\frac{18\!\cdots\!14}{11\!\cdots\!21}a^{11}-\frac{10\!\cdots\!00}{11\!\cdots\!21}a^{10}+\frac{16\!\cdots\!15}{11\!\cdots\!21}a^{9}-\frac{49\!\cdots\!44}{38\!\cdots\!91}a^{8}+\frac{76\!\cdots\!16}{11\!\cdots\!21}a^{7}-\frac{61\!\cdots\!94}{11\!\cdots\!21}a^{6}-\frac{12\!\cdots\!22}{11\!\cdots\!21}a^{5}+\frac{55\!\cdots\!13}{11\!\cdots\!21}a^{4}-\frac{14\!\cdots\!90}{11\!\cdots\!21}a^{3}-\frac{71\!\cdots\!58}{11\!\cdots\!21}a^{2}+\frac{26\!\cdots\!11}{11\!\cdots\!21}a-\frac{31\!\cdots\!87}{11\!\cdots\!21}$, $\frac{31\!\cdots\!82}{11\!\cdots\!21}a^{23}-\frac{23\!\cdots\!01}{11\!\cdots\!21}a^{22}+\frac{10\!\cdots\!75}{11\!\cdots\!21}a^{21}-\frac{29\!\cdots\!40}{11\!\cdots\!21}a^{20}+\frac{62\!\cdots\!33}{11\!\cdots\!21}a^{19}-\frac{95\!\cdots\!96}{11\!\cdots\!21}a^{18}+\frac{82\!\cdots\!50}{11\!\cdots\!21}a^{17}+\frac{57\!\cdots\!42}{11\!\cdots\!21}a^{16}-\frac{39\!\cdots\!82}{11\!\cdots\!21}a^{15}+\frac{87\!\cdots\!53}{11\!\cdots\!21}a^{14}-\frac{12\!\cdots\!82}{11\!\cdots\!21}a^{13}+\frac{94\!\cdots\!81}{11\!\cdots\!21}a^{12}+\frac{28\!\cdots\!49}{11\!\cdots\!21}a^{11}-\frac{20\!\cdots\!39}{11\!\cdots\!21}a^{10}+\frac{33\!\cdots\!86}{11\!\cdots\!21}a^{9}-\frac{10\!\cdots\!35}{38\!\cdots\!91}a^{8}+\frac{16\!\cdots\!76}{11\!\cdots\!21}a^{7}-\frac{20\!\cdots\!83}{11\!\cdots\!21}a^{6}-\frac{24\!\cdots\!18}{11\!\cdots\!21}a^{5}+\frac{12\!\cdots\!38}{11\!\cdots\!21}a^{4}-\frac{73\!\cdots\!23}{11\!\cdots\!21}a^{3}-\frac{14\!\cdots\!71}{11\!\cdots\!21}a^{2}+\frac{62\!\cdots\!18}{11\!\cdots\!21}a-\frac{27\!\cdots\!09}{40\!\cdots\!49}$, $\frac{49\!\cdots\!15}{11\!\cdots\!21}a^{23}-\frac{37\!\cdots\!00}{11\!\cdots\!21}a^{22}+\frac{15\!\cdots\!46}{11\!\cdots\!21}a^{21}-\frac{45\!\cdots\!32}{11\!\cdots\!21}a^{20}+\frac{95\!\cdots\!32}{11\!\cdots\!21}a^{19}-\frac{14\!\cdots\!40}{11\!\cdots\!21}a^{18}+\frac{11\!\cdots\!15}{11\!\cdots\!21}a^{17}+\frac{10\!\cdots\!19}{11\!\cdots\!21}a^{16}-\frac{21\!\cdots\!16}{40\!\cdots\!49}a^{15}+\frac{13\!\cdots\!58}{11\!\cdots\!21}a^{14}-\frac{18\!\cdots\!51}{11\!\cdots\!21}a^{13}+\frac{13\!\cdots\!51}{11\!\cdots\!21}a^{12}+\frac{55\!\cdots\!94}{11\!\cdots\!21}a^{11}-\frac{32\!\cdots\!41}{11\!\cdots\!21}a^{10}+\frac{50\!\cdots\!60}{11\!\cdots\!21}a^{9}-\frac{15\!\cdots\!27}{38\!\cdots\!91}a^{8}+\frac{23\!\cdots\!06}{11\!\cdots\!21}a^{7}-\frac{16\!\cdots\!49}{11\!\cdots\!21}a^{6}-\frac{38\!\cdots\!32}{11\!\cdots\!21}a^{5}+\frac{16\!\cdots\!86}{11\!\cdots\!21}a^{4}+\frac{30\!\cdots\!43}{11\!\cdots\!21}a^{3}-\frac{21\!\cdots\!26}{11\!\cdots\!21}a^{2}+\frac{74\!\cdots\!25}{11\!\cdots\!21}a-\frac{83\!\cdots\!21}{11\!\cdots\!21}$, $\frac{75\!\cdots\!59}{11\!\cdots\!21}a^{23}-\frac{57\!\cdots\!04}{11\!\cdots\!21}a^{22}+\frac{24\!\cdots\!49}{11\!\cdots\!21}a^{21}-\frac{69\!\cdots\!33}{11\!\cdots\!21}a^{20}+\frac{14\!\cdots\!77}{11\!\cdots\!21}a^{19}-\frac{22\!\cdots\!54}{11\!\cdots\!21}a^{18}+\frac{18\!\cdots\!94}{11\!\cdots\!21}a^{17}+\frac{14\!\cdots\!37}{11\!\cdots\!21}a^{16}-\frac{93\!\cdots\!19}{11\!\cdots\!21}a^{15}+\frac{20\!\cdots\!89}{11\!\cdots\!21}a^{14}-\frac{28\!\cdots\!46}{11\!\cdots\!21}a^{13}+\frac{21\!\cdots\!99}{11\!\cdots\!21}a^{12}+\frac{78\!\cdots\!47}{11\!\cdots\!21}a^{11}-\frac{49\!\cdots\!11}{11\!\cdots\!21}a^{10}+\frac{78\!\cdots\!89}{11\!\cdots\!21}a^{9}-\frac{23\!\cdots\!24}{38\!\cdots\!91}a^{8}+\frac{37\!\cdots\!14}{11\!\cdots\!21}a^{7}-\frac{33\!\cdots\!76}{11\!\cdots\!21}a^{6}-\frac{59\!\cdots\!46}{11\!\cdots\!21}a^{5}+\frac{27\!\cdots\!41}{11\!\cdots\!21}a^{4}-\frac{49\!\cdots\!81}{11\!\cdots\!21}a^{3}-\frac{35\!\cdots\!21}{11\!\cdots\!21}a^{2}+\frac{12\!\cdots\!33}{11\!\cdots\!21}a-\frac{14\!\cdots\!86}{11\!\cdots\!21}$, $\frac{96\!\cdots\!65}{11\!\cdots\!21}a^{23}-\frac{72\!\cdots\!18}{11\!\cdots\!21}a^{22}+\frac{30\!\cdots\!26}{11\!\cdots\!21}a^{21}-\frac{87\!\cdots\!55}{11\!\cdots\!21}a^{20}+\frac{18\!\cdots\!23}{11\!\cdots\!21}a^{19}-\frac{27\!\cdots\!36}{11\!\cdots\!21}a^{18}+\frac{23\!\cdots\!44}{11\!\cdots\!21}a^{17}+\frac{18\!\cdots\!43}{11\!\cdots\!21}a^{16}-\frac{11\!\cdots\!94}{11\!\cdots\!21}a^{15}+\frac{25\!\cdots\!59}{11\!\cdots\!21}a^{14}-\frac{35\!\cdots\!32}{11\!\cdots\!21}a^{13}+\frac{26\!\cdots\!61}{11\!\cdots\!21}a^{12}+\frac{10\!\cdots\!13}{11\!\cdots\!21}a^{11}-\frac{62\!\cdots\!17}{11\!\cdots\!21}a^{10}+\frac{98\!\cdots\!91}{11\!\cdots\!21}a^{9}-\frac{29\!\cdots\!55}{38\!\cdots\!91}a^{8}+\frac{45\!\cdots\!33}{11\!\cdots\!21}a^{7}-\frac{43\!\cdots\!15}{11\!\cdots\!21}a^{6}-\frac{71\!\cdots\!98}{11\!\cdots\!21}a^{5}+\frac{33\!\cdots\!43}{11\!\cdots\!21}a^{4}-\frac{12\!\cdots\!88}{11\!\cdots\!21}a^{3}-\frac{41\!\cdots\!09}{11\!\cdots\!21}a^{2}+\frac{16\!\cdots\!58}{11\!\cdots\!21}a-\frac{20\!\cdots\!02}{11\!\cdots\!21}$, $\frac{89\!\cdots\!12}{11\!\cdots\!21}a^{23}-\frac{70\!\cdots\!84}{11\!\cdots\!21}a^{22}+\frac{30\!\cdots\!44}{11\!\cdots\!21}a^{21}-\frac{89\!\cdots\!58}{11\!\cdots\!21}a^{20}+\frac{19\!\cdots\!86}{11\!\cdots\!21}a^{19}-\frac{30\!\cdots\!13}{11\!\cdots\!21}a^{18}+\frac{28\!\cdots\!52}{11\!\cdots\!21}a^{17}+\frac{12\!\cdots\!80}{11\!\cdots\!21}a^{16}-\frac{11\!\cdots\!39}{11\!\cdots\!21}a^{15}+\frac{27\!\cdots\!21}{11\!\cdots\!21}a^{14}-\frac{39\!\cdots\!46}{11\!\cdots\!21}a^{13}+\frac{33\!\cdots\!65}{11\!\cdots\!21}a^{12}+\frac{41\!\cdots\!36}{11\!\cdots\!21}a^{11}-\frac{62\!\cdots\!85}{11\!\cdots\!21}a^{10}+\frac{10\!\cdots\!54}{11\!\cdots\!21}a^{9}-\frac{35\!\cdots\!50}{38\!\cdots\!91}a^{8}+\frac{62\!\cdots\!34}{11\!\cdots\!21}a^{7}-\frac{11\!\cdots\!90}{11\!\cdots\!21}a^{6}-\frac{76\!\cdots\!64}{11\!\cdots\!21}a^{5}+\frac{48\!\cdots\!41}{11\!\cdots\!21}a^{4}-\frac{53\!\cdots\!57}{11\!\cdots\!21}a^{3}-\frac{48\!\cdots\!82}{11\!\cdots\!21}a^{2}+\frac{23\!\cdots\!64}{11\!\cdots\!21}a-\frac{36\!\cdots\!46}{11\!\cdots\!21}$
| 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: | \( 2176553.0653759805 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 2176553.0653759805 \cdot 1}{30\cdot\sqrt{1729054511370401309967041015625}}\cr\approx \mathstrut & 0.208882512574536 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1)
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^24 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1, 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^24 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1);
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^24 - 8*x^23 + 35*x^22 - 105*x^21 + 233*x^20 - 377*x^19 + 373*x^18 + 86*x^17 - 1310*x^16 + 3226*x^15 - 4893*x^14 + 4424*x^13 - 205*x^12 - 6944*x^11 + 13153*x^10 - 14094*x^9 + 9116*x^8 - 2660*x^7 - 525*x^6 + 689*x^5 - 175*x^4 - 37*x^3 + 37*x^2 - 10*x + 1);
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
$S_3\times C_{12}$ (as 24T65):
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 72 |
| The 36 conjugacy class representatives for $S_3\times C_{12}$ |
| Character table for $S_3\times C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.3243375.1, \(\Q(\zeta_{15})\), 12.0.10519481390625.2 |
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 36 siblings: | data not computed |
| 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 | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | R | R | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | R | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $24$ | $2$ | $12$ | $12$ | |||
|
\(5\)
| Deg $24$ | $4$ | $6$ | $18$ | |||
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.93.6t1.a.a | $1$ | $ 3 \cdot 31 $ | 6.0.24935067.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.31.3t1.a.a | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.93.6t1.a.b | $1$ | $ 3 \cdot 31 $ | 6.0.24935067.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.155.6t1.a.a | $1$ | $ 5 \cdot 31 $ | 6.6.115440125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.465.6t1.b.a | $1$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3116883375.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.155.6t1.a.b | $1$ | $ 5 \cdot 31 $ | 6.6.115440125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.465.6t1.b.b | $1$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3116883375.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.31.3t1.a.b | $1$ | $ 31 $ | 3.3.961.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.465.12t1.a.a | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.465.12t1.a.b | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.465.12t1.a.c | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.465.12t1.a.d | $1$ | $ 3 \cdot 5 \cdot 31 $ | 12.12.1214370246668923828125.1 | $C_{12}$ (as 12T1) | $0$ | $1$ | |
| 1.155.12t1.a.a | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.155.12t1.a.b | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.155.12t1.a.c | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.155.12t1.a.d | $1$ | $ 5 \cdot 31 $ | 12.0.1665802807501953125.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.14415.3t2.a.a | $2$ | $ 3 \cdot 5 \cdot 31^{2}$ | 3.1.14415.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.14415.6t3.e.a | $2$ | $ 3 \cdot 5 \cdot 31^{2}$ | 6.0.623376675.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.465.6t5.c.a | $2$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3243375.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.465.6t5.c.b | $2$ | $ 3 \cdot 5 \cdot 31 $ | 6.0.3243375.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.465.12t18.a.a | $2$ | $ 3 \cdot 5 \cdot 31 $ | 12.0.10519481390625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.465.12t18.a.b | $2$ | $ 3 \cdot 5 \cdot 31 $ | 12.0.10519481390625.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| 2.72075.12t11.b.a | $2$ | $ 3 \cdot 5^{2} \cdot 31^{2}$ | 12.0.134930027407658203125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| 2.72075.12t11.b.b | $2$ | $ 3 \cdot 5^{2} \cdot 31^{2}$ | 12.0.134930027407658203125.1 | $S_3 \times C_4$ (as 12T11) | $0$ | $0$ | |
| * | 2.2325.24t65.a.a | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2325.24t65.a.b | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2325.24t65.a.c | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
| * | 2.2325.24t65.a.d | $2$ | $ 3 \cdot 5^{2} \cdot 31 $ | 24.0.1729054511370401309967041015625.1 | $S_3\times C_{12}$ (as 24T65) | $0$ | $0$ |
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 *.