Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197)
gp: K = bnfinit(y^24 - 4*y^23 + 7*y^22 - 10*y^21 + 23*y^20 - 32*y^19 - 21*y^18 + 156*y^17 - 359*y^16 + 654*y^15 - 876*y^14 + 646*y^13 + 61*y^12 - 1012*y^11 + 2338*y^10 - 3178*y^9 + 2897*y^8 - 2474*y^7 + 1335*y^6 - 52*y^5 - 44*y^4 + 454*y^3 - 524*y^2 + 62*y - 197, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197)
\( x^{24} - 4 x^{23} + 7 x^{22} - 10 x^{21} + 23 x^{20} - 32 x^{19} - 21 x^{18} + 156 x^{17} - 359 x^{16} + \cdots - 197 \)
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: | $[4, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(49177850545349555386638457176064\)
\(\medspace = 2^{16}\cdot 487^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.92\) | 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: | $2^{2/3}487^{1/2}\approx 35.030887836273955$ | ||
| Ramified primes: |
\(2\), \(487\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{11\!\cdots\!82}a^{23}+\frac{13\!\cdots\!41}{11\!\cdots\!82}a^{22}-\frac{11\!\cdots\!21}{11\!\cdots\!82}a^{21}+\frac{23\!\cdots\!29}{11\!\cdots\!82}a^{20}-\frac{75\!\cdots\!55}{57\!\cdots\!41}a^{19}-\frac{99\!\cdots\!86}{57\!\cdots\!41}a^{18}-\frac{16\!\cdots\!13}{11\!\cdots\!82}a^{17}-\frac{26\!\cdots\!97}{11\!\cdots\!82}a^{16}+\frac{24\!\cdots\!13}{11\!\cdots\!82}a^{15}+\frac{87\!\cdots\!18}{57\!\cdots\!41}a^{14}+\frac{56\!\cdots\!11}{11\!\cdots\!82}a^{13}+\frac{11\!\cdots\!74}{57\!\cdots\!41}a^{12}-\frac{13\!\cdots\!42}{57\!\cdots\!41}a^{11}+\frac{17\!\cdots\!87}{57\!\cdots\!41}a^{10}+\frac{20\!\cdots\!87}{11\!\cdots\!82}a^{9}-\frac{26\!\cdots\!14}{57\!\cdots\!41}a^{8}-\frac{18\!\cdots\!21}{11\!\cdots\!82}a^{7}-\frac{56\!\cdots\!27}{11\!\cdots\!82}a^{6}+\frac{67\!\cdots\!28}{57\!\cdots\!41}a^{5}-\frac{26\!\cdots\!20}{57\!\cdots\!41}a^{4}+\frac{22\!\cdots\!54}{57\!\cdots\!41}a^{3}-\frac{21\!\cdots\!83}{57\!\cdots\!41}a^{2}+\frac{12\!\cdots\!60}{57\!\cdots\!41}a+\frac{38\!\cdots\!25}{11\!\cdots\!82}$
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: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$\frac{46\!\cdots\!96}{57\!\cdots\!41}a^{23}-\frac{39\!\cdots\!31}{11\!\cdots\!82}a^{22}+\frac{46\!\cdots\!51}{57\!\cdots\!41}a^{21}-\frac{92\!\cdots\!71}{57\!\cdots\!41}a^{20}+\frac{18\!\cdots\!24}{57\!\cdots\!41}a^{19}-\frac{46\!\cdots\!93}{11\!\cdots\!82}a^{18}+\frac{12\!\cdots\!29}{11\!\cdots\!82}a^{17}+\frac{11\!\cdots\!71}{11\!\cdots\!82}a^{16}-\frac{21\!\cdots\!68}{57\!\cdots\!41}a^{15}+\frac{52\!\cdots\!35}{57\!\cdots\!41}a^{14}-\frac{17\!\cdots\!33}{11\!\cdots\!82}a^{13}+\frac{17\!\cdots\!67}{11\!\cdots\!82}a^{12}-\frac{38\!\cdots\!41}{57\!\cdots\!41}a^{11}-\frac{70\!\cdots\!44}{57\!\cdots\!41}a^{10}+\frac{22\!\cdots\!52}{57\!\cdots\!41}a^{9}-\frac{34\!\cdots\!73}{57\!\cdots\!41}a^{8}+\frac{36\!\cdots\!86}{57\!\cdots\!41}a^{7}-\frac{51\!\cdots\!01}{11\!\cdots\!82}a^{6}+\frac{43\!\cdots\!47}{57\!\cdots\!41}a^{5}+\frac{10\!\cdots\!52}{57\!\cdots\!41}a^{4}-\frac{13\!\cdots\!22}{57\!\cdots\!41}a^{3}+\frac{12\!\cdots\!10}{57\!\cdots\!41}a^{2}-\frac{46\!\cdots\!47}{11\!\cdots\!82}a+\frac{62\!\cdots\!89}{11\!\cdots\!82}$, $\frac{16\!\cdots\!59}{11\!\cdots\!82}a^{23}+\frac{13\!\cdots\!34}{57\!\cdots\!41}a^{22}-\frac{14\!\cdots\!61}{57\!\cdots\!41}a^{21}+\frac{28\!\cdots\!84}{57\!\cdots\!41}a^{20}-\frac{61\!\cdots\!53}{11\!\cdots\!82}a^{19}+\frac{83\!\cdots\!39}{57\!\cdots\!41}a^{18}-\frac{18\!\cdots\!92}{57\!\cdots\!41}a^{17}+\frac{42\!\cdots\!72}{57\!\cdots\!41}a^{16}+\frac{10\!\cdots\!65}{11\!\cdots\!82}a^{15}-\frac{13\!\cdots\!73}{57\!\cdots\!41}a^{14}+\frac{51\!\cdots\!91}{11\!\cdots\!82}a^{13}-\frac{39\!\cdots\!56}{57\!\cdots\!41}a^{12}+\frac{32\!\cdots\!98}{57\!\cdots\!41}a^{11}-\frac{72\!\cdots\!01}{11\!\cdots\!82}a^{10}-\frac{80\!\cdots\!27}{11\!\cdots\!82}a^{9}+\frac{96\!\cdots\!94}{57\!\cdots\!41}a^{8}-\frac{28\!\cdots\!11}{11\!\cdots\!82}a^{7}+\frac{24\!\cdots\!99}{11\!\cdots\!82}a^{6}-\frac{92\!\cdots\!51}{57\!\cdots\!41}a^{5}+\frac{48\!\cdots\!90}{57\!\cdots\!41}a^{4}+\frac{15\!\cdots\!09}{57\!\cdots\!41}a^{3}-\frac{32\!\cdots\!21}{11\!\cdots\!82}a^{2}+\frac{24\!\cdots\!17}{11\!\cdots\!82}a-\frac{37\!\cdots\!67}{11\!\cdots\!82}$, $\frac{11\!\cdots\!71}{11\!\cdots\!82}a^{23}-\frac{63\!\cdots\!15}{11\!\cdots\!82}a^{22}+\frac{12\!\cdots\!17}{11\!\cdots\!82}a^{21}-\frac{13\!\cdots\!03}{11\!\cdots\!82}a^{20}+\frac{14\!\cdots\!50}{57\!\cdots\!41}a^{19}-\frac{57\!\cdots\!13}{11\!\cdots\!82}a^{18}-\frac{88\!\cdots\!62}{57\!\cdots\!41}a^{17}+\frac{27\!\cdots\!17}{11\!\cdots\!82}a^{16}-\frac{29\!\cdots\!91}{57\!\cdots\!41}a^{15}+\frac{47\!\cdots\!15}{57\!\cdots\!41}a^{14}-\frac{12\!\cdots\!91}{11\!\cdots\!82}a^{13}+\frac{91\!\cdots\!01}{11\!\cdots\!82}a^{12}+\frac{45\!\cdots\!91}{11\!\cdots\!82}a^{11}-\frac{90\!\cdots\!41}{57\!\cdots\!41}a^{10}+\frac{15\!\cdots\!58}{57\!\cdots\!41}a^{9}-\frac{22\!\cdots\!49}{57\!\cdots\!41}a^{8}+\frac{18\!\cdots\!27}{57\!\cdots\!41}a^{7}-\frac{11\!\cdots\!11}{57\!\cdots\!41}a^{6}+\frac{10\!\cdots\!09}{57\!\cdots\!41}a^{5}-\frac{60\!\cdots\!14}{57\!\cdots\!41}a^{4}+\frac{90\!\cdots\!19}{11\!\cdots\!82}a^{3}-\frac{77\!\cdots\!51}{11\!\cdots\!82}a^{2}+\frac{92\!\cdots\!23}{11\!\cdots\!82}a+\frac{32\!\cdots\!29}{11\!\cdots\!82}$, $\frac{89\!\cdots\!35}{11\!\cdots\!82}a^{23}-\frac{38\!\cdots\!41}{11\!\cdots\!82}a^{22}+\frac{65\!\cdots\!69}{11\!\cdots\!82}a^{21}-\frac{84\!\cdots\!79}{11\!\cdots\!82}a^{20}+\frac{10\!\cdots\!18}{57\!\cdots\!41}a^{19}-\frac{30\!\cdots\!55}{11\!\cdots\!82}a^{18}-\frac{11\!\cdots\!43}{57\!\cdots\!41}a^{17}+\frac{78\!\cdots\!50}{57\!\cdots\!41}a^{16}-\frac{16\!\cdots\!64}{57\!\cdots\!41}a^{15}+\frac{28\!\cdots\!76}{57\!\cdots\!41}a^{14}-\frac{77\!\cdots\!25}{11\!\cdots\!82}a^{13}+\frac{25\!\cdots\!05}{57\!\cdots\!41}a^{12}+\frac{23\!\cdots\!99}{11\!\cdots\!82}a^{11}-\frac{51\!\cdots\!03}{57\!\cdots\!41}a^{10}+\frac{10\!\cdots\!39}{57\!\cdots\!41}a^{9}-\frac{14\!\cdots\!79}{57\!\cdots\!41}a^{8}+\frac{11\!\cdots\!18}{57\!\cdots\!41}a^{7}-\frac{84\!\cdots\!86}{57\!\cdots\!41}a^{6}+\frac{47\!\cdots\!08}{57\!\cdots\!41}a^{5}+\frac{11\!\cdots\!66}{57\!\cdots\!41}a^{4}-\frac{27\!\cdots\!01}{11\!\cdots\!82}a^{3}+\frac{16\!\cdots\!89}{11\!\cdots\!82}a^{2}-\frac{42\!\cdots\!27}{11\!\cdots\!82}a+\frac{33\!\cdots\!30}{57\!\cdots\!41}$, $\frac{85\!\cdots\!03}{57\!\cdots\!41}a^{23}-\frac{35\!\cdots\!81}{57\!\cdots\!41}a^{22}+\frac{11\!\cdots\!35}{11\!\cdots\!82}a^{21}-\frac{14\!\cdots\!67}{11\!\cdots\!82}a^{20}+\frac{17\!\cdots\!39}{57\!\cdots\!41}a^{19}-\frac{50\!\cdots\!55}{11\!\cdots\!82}a^{18}-\frac{25\!\cdots\!18}{57\!\cdots\!41}a^{17}+\frac{15\!\cdots\!29}{57\!\cdots\!41}a^{16}-\frac{62\!\cdots\!87}{11\!\cdots\!82}a^{15}+\frac{10\!\cdots\!59}{11\!\cdots\!82}a^{14}-\frac{62\!\cdots\!97}{57\!\cdots\!41}a^{13}+\frac{64\!\cdots\!81}{11\!\cdots\!82}a^{12}+\frac{38\!\cdots\!19}{57\!\cdots\!41}a^{11}-\frac{22\!\cdots\!29}{11\!\cdots\!82}a^{10}+\frac{38\!\cdots\!61}{11\!\cdots\!82}a^{9}-\frac{23\!\cdots\!46}{57\!\cdots\!41}a^{8}+\frac{15\!\cdots\!72}{57\!\cdots\!41}a^{7}-\frac{19\!\cdots\!99}{11\!\cdots\!82}a^{6}+\frac{43\!\cdots\!25}{11\!\cdots\!82}a^{5}+\frac{11\!\cdots\!31}{11\!\cdots\!82}a^{4}-\frac{27\!\cdots\!46}{57\!\cdots\!41}a^{3}+\frac{21\!\cdots\!63}{57\!\cdots\!41}a^{2}-\frac{27\!\cdots\!24}{57\!\cdots\!41}a-\frac{21\!\cdots\!11}{57\!\cdots\!41}$, $\frac{46\!\cdots\!96}{57\!\cdots\!41}a^{23}-\frac{39\!\cdots\!31}{11\!\cdots\!82}a^{22}+\frac{46\!\cdots\!51}{57\!\cdots\!41}a^{21}-\frac{92\!\cdots\!71}{57\!\cdots\!41}a^{20}+\frac{18\!\cdots\!24}{57\!\cdots\!41}a^{19}-\frac{46\!\cdots\!93}{11\!\cdots\!82}a^{18}+\frac{12\!\cdots\!29}{11\!\cdots\!82}a^{17}+\frac{11\!\cdots\!71}{11\!\cdots\!82}a^{16}-\frac{21\!\cdots\!68}{57\!\cdots\!41}a^{15}+\frac{52\!\cdots\!35}{57\!\cdots\!41}a^{14}-\frac{17\!\cdots\!33}{11\!\cdots\!82}a^{13}+\frac{17\!\cdots\!67}{11\!\cdots\!82}a^{12}-\frac{38\!\cdots\!41}{57\!\cdots\!41}a^{11}-\frac{70\!\cdots\!44}{57\!\cdots\!41}a^{10}+\frac{22\!\cdots\!52}{57\!\cdots\!41}a^{9}-\frac{34\!\cdots\!73}{57\!\cdots\!41}a^{8}+\frac{36\!\cdots\!86}{57\!\cdots\!41}a^{7}-\frac{51\!\cdots\!01}{11\!\cdots\!82}a^{6}+\frac{43\!\cdots\!47}{57\!\cdots\!41}a^{5}+\frac{10\!\cdots\!52}{57\!\cdots\!41}a^{4}-\frac{13\!\cdots\!22}{57\!\cdots\!41}a^{3}+\frac{12\!\cdots\!10}{57\!\cdots\!41}a^{2}-\frac{46\!\cdots\!47}{11\!\cdots\!82}a+\frac{12\!\cdots\!71}{11\!\cdots\!82}$, $\frac{43\!\cdots\!79}{11\!\cdots\!82}a^{23}-\frac{99\!\cdots\!91}{11\!\cdots\!82}a^{22}+\frac{94\!\cdots\!31}{11\!\cdots\!82}a^{21}-\frac{21\!\cdots\!63}{11\!\cdots\!82}a^{20}+\frac{34\!\cdots\!15}{57\!\cdots\!41}a^{19}-\frac{15\!\cdots\!55}{57\!\cdots\!41}a^{18}-\frac{16\!\cdots\!61}{11\!\cdots\!82}a^{17}+\frac{34\!\cdots\!73}{11\!\cdots\!82}a^{16}-\frac{69\!\cdots\!61}{11\!\cdots\!82}a^{15}+\frac{68\!\cdots\!72}{57\!\cdots\!41}a^{14}-\frac{14\!\cdots\!17}{11\!\cdots\!82}a^{13}+\frac{33\!\cdots\!06}{57\!\cdots\!41}a^{12}-\frac{11\!\cdots\!74}{57\!\cdots\!41}a^{11}-\frac{63\!\cdots\!92}{57\!\cdots\!41}a^{10}+\frac{51\!\cdots\!21}{11\!\cdots\!82}a^{9}-\frac{24\!\cdots\!92}{57\!\cdots\!41}a^{8}+\frac{55\!\cdots\!93}{11\!\cdots\!82}a^{7}-\frac{81\!\cdots\!47}{11\!\cdots\!82}a^{6}+\frac{11\!\cdots\!14}{57\!\cdots\!41}a^{5}-\frac{59\!\cdots\!59}{57\!\cdots\!41}a^{4}+\frac{19\!\cdots\!92}{57\!\cdots\!41}a^{3}+\frac{18\!\cdots\!16}{57\!\cdots\!41}a^{2}-\frac{32\!\cdots\!22}{57\!\cdots\!41}a+\frac{11\!\cdots\!17}{11\!\cdots\!82}$, $\frac{23\!\cdots\!81}{57\!\cdots\!41}a^{23}+\frac{13\!\cdots\!99}{57\!\cdots\!41}a^{22}-\frac{91\!\cdots\!71}{11\!\cdots\!82}a^{21}+\frac{85\!\cdots\!71}{11\!\cdots\!82}a^{20}-\frac{13\!\cdots\!59}{11\!\cdots\!82}a^{19}+\frac{34\!\cdots\!15}{57\!\cdots\!41}a^{18}-\frac{36\!\cdots\!42}{57\!\cdots\!41}a^{17}-\frac{53\!\cdots\!31}{57\!\cdots\!41}a^{16}+\frac{27\!\cdots\!75}{11\!\cdots\!82}a^{15}-\frac{49\!\cdots\!67}{11\!\cdots\!82}a^{14}+\frac{12\!\cdots\!25}{11\!\cdots\!82}a^{13}-\frac{18\!\cdots\!23}{11\!\cdots\!82}a^{12}+\frac{62\!\cdots\!45}{57\!\cdots\!41}a^{11}-\frac{26\!\cdots\!85}{57\!\cdots\!41}a^{10}-\frac{44\!\cdots\!31}{11\!\cdots\!82}a^{9}+\frac{44\!\cdots\!67}{11\!\cdots\!82}a^{8}-\frac{60\!\cdots\!19}{11\!\cdots\!82}a^{7}+\frac{61\!\cdots\!91}{11\!\cdots\!82}a^{6}-\frac{82\!\cdots\!75}{11\!\cdots\!82}a^{5}+\frac{19\!\cdots\!95}{57\!\cdots\!41}a^{4}-\frac{11\!\cdots\!23}{11\!\cdots\!82}a^{3}+\frac{16\!\cdots\!43}{11\!\cdots\!82}a^{2}+\frac{56\!\cdots\!28}{57\!\cdots\!41}a-\frac{76\!\cdots\!57}{11\!\cdots\!82}$, $\frac{48\!\cdots\!16}{57\!\cdots\!41}a^{23}-\frac{20\!\cdots\!47}{57\!\cdots\!41}a^{22}+\frac{66\!\cdots\!71}{11\!\cdots\!82}a^{21}-\frac{74\!\cdots\!65}{11\!\cdots\!82}a^{20}+\frac{91\!\cdots\!99}{57\!\cdots\!41}a^{19}-\frac{28\!\cdots\!33}{11\!\cdots\!82}a^{18}-\frac{16\!\cdots\!14}{57\!\cdots\!41}a^{17}+\frac{18\!\cdots\!79}{11\!\cdots\!82}a^{16}-\frac{35\!\cdots\!71}{11\!\cdots\!82}a^{15}+\frac{27\!\cdots\!81}{57\!\cdots\!41}a^{14}-\frac{33\!\cdots\!18}{57\!\cdots\!41}a^{13}+\frac{26\!\cdots\!71}{11\!\cdots\!82}a^{12}+\frac{59\!\cdots\!73}{11\!\cdots\!82}a^{11}-\frac{68\!\cdots\!10}{57\!\cdots\!41}a^{10}+\frac{21\!\cdots\!35}{11\!\cdots\!82}a^{9}-\frac{12\!\cdots\!74}{57\!\cdots\!41}a^{8}+\frac{14\!\cdots\!83}{11\!\cdots\!82}a^{7}-\frac{55\!\cdots\!93}{11\!\cdots\!82}a^{6}-\frac{25\!\cdots\!11}{11\!\cdots\!82}a^{5}+\frac{94\!\cdots\!09}{11\!\cdots\!82}a^{4}-\frac{32\!\cdots\!05}{11\!\cdots\!82}a^{3}+\frac{94\!\cdots\!41}{11\!\cdots\!82}a^{2}-\frac{36\!\cdots\!71}{11\!\cdots\!82}a-\frac{57\!\cdots\!40}{57\!\cdots\!41}$, $\frac{85\!\cdots\!05}{11\!\cdots\!82}a^{23}-\frac{16\!\cdots\!85}{57\!\cdots\!41}a^{22}+\frac{53\!\cdots\!95}{11\!\cdots\!82}a^{21}-\frac{32\!\cdots\!95}{57\!\cdots\!41}a^{20}+\frac{80\!\cdots\!90}{57\!\cdots\!41}a^{19}-\frac{22\!\cdots\!69}{11\!\cdots\!82}a^{18}-\frac{26\!\cdots\!37}{11\!\cdots\!82}a^{17}+\frac{14\!\cdots\!91}{11\!\cdots\!82}a^{16}-\frac{28\!\cdots\!15}{11\!\cdots\!82}a^{15}+\frac{46\!\cdots\!09}{11\!\cdots\!82}a^{14}-\frac{55\!\cdots\!65}{11\!\cdots\!82}a^{13}+\frac{26\!\cdots\!73}{11\!\cdots\!82}a^{12}+\frac{18\!\cdots\!04}{57\!\cdots\!41}a^{11}-\frac{50\!\cdots\!23}{57\!\cdots\!41}a^{10}+\frac{88\!\cdots\!08}{57\!\cdots\!41}a^{9}-\frac{20\!\cdots\!35}{11\!\cdots\!82}a^{8}+\frac{75\!\cdots\!16}{57\!\cdots\!41}a^{7}-\frac{87\!\cdots\!09}{11\!\cdots\!82}a^{6}+\frac{89\!\cdots\!85}{57\!\cdots\!41}a^{5}+\frac{27\!\cdots\!91}{57\!\cdots\!41}a^{4}-\frac{18\!\cdots\!93}{57\!\cdots\!41}a^{3}+\frac{25\!\cdots\!47}{11\!\cdots\!82}a^{2}-\frac{28\!\cdots\!93}{11\!\cdots\!82}a-\frac{16\!\cdots\!27}{57\!\cdots\!41}$, $\frac{73\!\cdots\!39}{11\!\cdots\!82}a^{23}-\frac{20\!\cdots\!73}{11\!\cdots\!82}a^{22}+\frac{85\!\cdots\!93}{11\!\cdots\!82}a^{21}+\frac{53\!\cdots\!90}{57\!\cdots\!41}a^{20}+\frac{55\!\cdots\!95}{11\!\cdots\!82}a^{19}-\frac{56\!\cdots\!61}{11\!\cdots\!82}a^{18}-\frac{53\!\cdots\!55}{11\!\cdots\!82}a^{17}+\frac{51\!\cdots\!14}{57\!\cdots\!41}a^{16}-\frac{44\!\cdots\!36}{57\!\cdots\!41}a^{15}+\frac{30\!\cdots\!09}{57\!\cdots\!41}a^{14}+\frac{89\!\cdots\!03}{11\!\cdots\!82}a^{13}-\frac{26\!\cdots\!18}{57\!\cdots\!41}a^{12}+\frac{40\!\cdots\!12}{57\!\cdots\!41}a^{11}-\frac{29\!\cdots\!10}{57\!\cdots\!41}a^{10}+\frac{17\!\cdots\!08}{57\!\cdots\!41}a^{9}+\frac{37\!\cdots\!27}{11\!\cdots\!82}a^{8}-\frac{78\!\cdots\!51}{57\!\cdots\!41}a^{7}+\frac{71\!\cdots\!29}{57\!\cdots\!41}a^{6}-\frac{60\!\cdots\!43}{57\!\cdots\!41}a^{5}+\frac{48\!\cdots\!96}{57\!\cdots\!41}a^{4}+\frac{29\!\cdots\!71}{11\!\cdots\!82}a^{3}-\frac{11\!\cdots\!32}{57\!\cdots\!41}a^{2}-\frac{45\!\cdots\!61}{11\!\cdots\!82}a-\frac{42\!\cdots\!95}{11\!\cdots\!82}$, $\frac{13\!\cdots\!39}{11\!\cdots\!82}a^{23}-\frac{30\!\cdots\!62}{57\!\cdots\!41}a^{22}+\frac{12\!\cdots\!85}{11\!\cdots\!82}a^{21}-\frac{18\!\cdots\!31}{11\!\cdots\!82}a^{20}+\frac{18\!\cdots\!79}{57\!\cdots\!41}a^{19}-\frac{29\!\cdots\!60}{57\!\cdots\!41}a^{18}-\frac{30\!\cdots\!14}{57\!\cdots\!41}a^{17}+\frac{11\!\cdots\!72}{57\!\cdots\!41}a^{16}-\frac{62\!\cdots\!19}{11\!\cdots\!82}a^{15}+\frac{11\!\cdots\!53}{11\!\cdots\!82}a^{14}-\frac{16\!\cdots\!27}{11\!\cdots\!82}a^{13}+\frac{69\!\cdots\!66}{57\!\cdots\!41}a^{12}-\frac{11\!\cdots\!09}{11\!\cdots\!82}a^{11}-\frac{18\!\cdots\!65}{11\!\cdots\!82}a^{10}+\frac{21\!\cdots\!40}{57\!\cdots\!41}a^{9}-\frac{30\!\cdots\!59}{57\!\cdots\!41}a^{8}+\frac{29\!\cdots\!54}{57\!\cdots\!41}a^{7}-\frac{22\!\cdots\!10}{57\!\cdots\!41}a^{6}+\frac{21\!\cdots\!17}{11\!\cdots\!82}a^{5}+\frac{29\!\cdots\!15}{11\!\cdots\!82}a^{4}-\frac{58\!\cdots\!35}{57\!\cdots\!41}a^{3}+\frac{10\!\cdots\!93}{11\!\cdots\!82}a^{2}-\frac{41\!\cdots\!13}{57\!\cdots\!41}a+\frac{15\!\cdots\!73}{57\!\cdots\!41}$, $\frac{50\!\cdots\!63}{11\!\cdots\!82}a^{23}-\frac{19\!\cdots\!35}{11\!\cdots\!82}a^{22}+\frac{16\!\cdots\!43}{57\!\cdots\!41}a^{21}-\frac{46\!\cdots\!07}{11\!\cdots\!82}a^{20}+\frac{10\!\cdots\!09}{11\!\cdots\!82}a^{19}-\frac{13\!\cdots\!69}{11\!\cdots\!82}a^{18}-\frac{12\!\cdots\!71}{11\!\cdots\!82}a^{17}+\frac{39\!\cdots\!24}{57\!\cdots\!41}a^{16}-\frac{89\!\cdots\!55}{57\!\cdots\!41}a^{15}+\frac{30\!\cdots\!63}{11\!\cdots\!82}a^{14}-\frac{38\!\cdots\!25}{11\!\cdots\!82}a^{13}+\frac{12\!\cdots\!40}{57\!\cdots\!41}a^{12}+\frac{14\!\cdots\!87}{11\!\cdots\!82}a^{11}-\frac{63\!\cdots\!11}{11\!\cdots\!82}a^{10}+\frac{59\!\cdots\!05}{57\!\cdots\!41}a^{9}-\frac{71\!\cdots\!64}{57\!\cdots\!41}a^{8}+\frac{60\!\cdots\!37}{57\!\cdots\!41}a^{7}-\frac{72\!\cdots\!37}{11\!\cdots\!82}a^{6}+\frac{58\!\cdots\!05}{57\!\cdots\!41}a^{5}+\frac{22\!\cdots\!03}{11\!\cdots\!82}a^{4}-\frac{37\!\cdots\!61}{11\!\cdots\!82}a^{3}+\frac{21\!\cdots\!67}{11\!\cdots\!82}a^{2}-\frac{59\!\cdots\!76}{57\!\cdots\!41}a+\frac{62\!\cdots\!15}{57\!\cdots\!41}$
| 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: | \( 1432261.5638336123 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 1432261.5638336123 \cdot 1}{2\cdot\sqrt{49177850545349555386638457176064}}\cr\approx \mathstrut & 0.156684572333305 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197)
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 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197, 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 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197);
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 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197);
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
$\SL(2,5):C_2$ (as 24T576):
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 non-solvable group of order 240 |
| The 18 conjugacy class representatives for $\SL(2,5):C_2$ |
| Character table for $\SL(2,5):C_2$ |
Intermediate fields
| 6.2.3794704.1, 12.4.14399778447616.1 |
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 40 siblings: | data not computed |
| Arithmetically equvalently sibling: | 24.4.49177850545349555386638457176064.1 |
| 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 | R | $20{,}\,{\href{/padicField/3.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }^{6}$ | $20{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }^{6}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(487\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.487.2t1.a.a | $1$ | $ 487 $ | \(\Q(\sqrt{-487}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.1948.120.a.a | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.1948.120.a.b | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.1948.120.a.c | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.1948.120.a.d | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| * | 3.1948.12t76.a.a | $3$ | $ 2^{2} \cdot 487 $ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| * | 3.1948.12t76.a.b | $3$ | $ 2^{2} \cdot 487 $ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| 3.948676.12t33.a.a | $3$ | $ 2^{2} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 3.948676.12t33.a.b | $3$ | $ 2^{2} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 4.948676.10t11.a.a | $4$ | $ 2^{2} \cdot 487^{2}$ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
| 4.948676.5t4.a.a | $4$ | $ 2^{2} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
| 4.948676.40t188.a.a | $4$ | $ 2^{2} \cdot 487^{2}$ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 4.948676.40t188.a.b | $4$ | $ 2^{2} \cdot 487^{2}$ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 5.1848020848.12t75.a.a | $5$ | $ 2^{4} \cdot 487^{3}$ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
| * | 5.3794704.6t12.a.a | $5$ | $ 2^{4} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $1$ |
| * | 6.1848020848.24t576.a.a | $6$ | $ 2^{4} \cdot 487^{3}$ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
| * | 6.1848020848.24t576.a.b | $6$ | $ 2^{4} \cdot 487^{3}$ | 24.4.49177850545349555386638457176064.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
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 *.