Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*x + 1)
gp: K = bnfinit(y^24 - 10*y^23 + 48*y^22 - 158*y^21 + 381*y^20 - 714*y^19 + 1213*y^18 - 1907*y^17 + 2614*y^16 - 3516*y^15 + 4694*y^14 - 5466*y^13 + 6009*y^12 - 6785*y^11 + 7506*y^10 - 8491*y^9 + 9365*y^8 - 8498*y^7 + 5899*y^6 - 3118*y^5 + 1203*y^4 - 296*y^3 + 36*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*x + 1)
\( x^{24} - 10 x^{23} + 48 x^{22} - 158 x^{21} + 381 x^{20} - 714 x^{19} + 1213 x^{18} - 1907 x^{17} + \cdots + 1 \)
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: | $[4, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(940350029043078386535797119140625\)
\(\medspace = 5^{16}\cdot 151^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.65\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $5^{2/3}151^{1/2}\approx 35.93093151785668$ | ||
| Ramified primes: |
\(5\), \(151\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{87\!\cdots\!72}a^{23}+\frac{29\!\cdots\!49}{87\!\cdots\!72}a^{22}+\frac{14\!\cdots\!59}{87\!\cdots\!72}a^{21}-\frac{78\!\cdots\!05}{87\!\cdots\!72}a^{20}+\frac{21\!\cdots\!83}{43\!\cdots\!86}a^{19}+\frac{23\!\cdots\!61}{21\!\cdots\!43}a^{18}+\frac{28\!\cdots\!77}{87\!\cdots\!72}a^{17}+\frac{99\!\cdots\!38}{21\!\cdots\!43}a^{16}-\frac{20\!\cdots\!85}{43\!\cdots\!86}a^{15}-\frac{11\!\cdots\!15}{43\!\cdots\!86}a^{14}+\frac{66\!\cdots\!89}{21\!\cdots\!43}a^{13}+\frac{12\!\cdots\!17}{43\!\cdots\!86}a^{12}-\frac{39\!\cdots\!33}{87\!\cdots\!72}a^{11}+\frac{72\!\cdots\!39}{21\!\cdots\!43}a^{10}-\frac{43\!\cdots\!15}{43\!\cdots\!86}a^{9}-\frac{29\!\cdots\!57}{87\!\cdots\!72}a^{8}+\frac{54\!\cdots\!53}{43\!\cdots\!86}a^{7}-\frac{44\!\cdots\!54}{21\!\cdots\!43}a^{6}+\frac{19\!\cdots\!67}{87\!\cdots\!72}a^{5}-\frac{73\!\cdots\!93}{87\!\cdots\!72}a^{4}+\frac{94\!\cdots\!07}{21\!\cdots\!43}a^{3}-\frac{92\!\cdots\!77}{21\!\cdots\!43}a^{2}-\frac{10\!\cdots\!23}{21\!\cdots\!43}a-\frac{28\!\cdots\!11}{87\!\cdots\!72}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$\frac{21\!\cdots\!99}{21\!\cdots\!43}a^{23}-\frac{20\!\cdots\!64}{21\!\cdots\!43}a^{22}+\frac{94\!\cdots\!26}{21\!\cdots\!43}a^{21}-\frac{30\!\cdots\!51}{21\!\cdots\!43}a^{20}+\frac{70\!\cdots\!89}{21\!\cdots\!43}a^{19}-\frac{12\!\cdots\!81}{21\!\cdots\!43}a^{18}+\frac{22\!\cdots\!14}{21\!\cdots\!43}a^{17}-\frac{34\!\cdots\!83}{21\!\cdots\!43}a^{16}+\frac{46\!\cdots\!31}{21\!\cdots\!43}a^{15}-\frac{63\!\cdots\!18}{21\!\cdots\!43}a^{14}+\frac{82\!\cdots\!74}{21\!\cdots\!43}a^{13}-\frac{94\!\cdots\!19}{21\!\cdots\!43}a^{12}+\frac{10\!\cdots\!45}{21\!\cdots\!43}a^{11}-\frac{11\!\cdots\!82}{21\!\cdots\!43}a^{10}+\frac{13\!\cdots\!82}{21\!\cdots\!43}a^{9}-\frac{14\!\cdots\!59}{21\!\cdots\!43}a^{8}+\frac{16\!\cdots\!81}{21\!\cdots\!43}a^{7}-\frac{14\!\cdots\!22}{21\!\cdots\!43}a^{6}+\frac{98\!\cdots\!63}{21\!\cdots\!43}a^{5}-\frac{51\!\cdots\!19}{21\!\cdots\!43}a^{4}+\frac{19\!\cdots\!64}{21\!\cdots\!43}a^{3}-\frac{38\!\cdots\!77}{21\!\cdots\!43}a^{2}-\frac{10\!\cdots\!50}{21\!\cdots\!43}a-\frac{62\!\cdots\!42}{21\!\cdots\!43}$, $\frac{69\!\cdots\!07}{55\!\cdots\!49}a^{23}-\frac{70\!\cdots\!46}{55\!\cdots\!49}a^{22}+\frac{34\!\cdots\!36}{55\!\cdots\!49}a^{21}-\frac{11\!\cdots\!03}{55\!\cdots\!49}a^{20}+\frac{27\!\cdots\!71}{55\!\cdots\!49}a^{19}-\frac{52\!\cdots\!31}{55\!\cdots\!49}a^{18}+\frac{89\!\cdots\!94}{55\!\cdots\!49}a^{17}-\frac{14\!\cdots\!28}{55\!\cdots\!49}a^{16}+\frac{19\!\cdots\!16}{55\!\cdots\!49}a^{15}-\frac{25\!\cdots\!06}{55\!\cdots\!49}a^{14}+\frac{34\!\cdots\!15}{55\!\cdots\!49}a^{13}-\frac{40\!\cdots\!29}{55\!\cdots\!49}a^{12}+\frac{44\!\cdots\!03}{55\!\cdots\!49}a^{11}-\frac{50\!\cdots\!33}{55\!\cdots\!49}a^{10}+\frac{55\!\cdots\!12}{55\!\cdots\!49}a^{9}-\frac{62\!\cdots\!66}{55\!\cdots\!49}a^{8}+\frac{69\!\cdots\!49}{55\!\cdots\!49}a^{7}-\frac{63\!\cdots\!16}{55\!\cdots\!49}a^{6}+\frac{44\!\cdots\!08}{55\!\cdots\!49}a^{5}-\frac{23\!\cdots\!45}{55\!\cdots\!49}a^{4}+\frac{91\!\cdots\!34}{55\!\cdots\!49}a^{3}-\frac{22\!\cdots\!93}{55\!\cdots\!49}a^{2}+\frac{32\!\cdots\!77}{55\!\cdots\!49}a-\frac{98\!\cdots\!13}{55\!\cdots\!49}$, $\frac{14\!\cdots\!96}{21\!\cdots\!43}a^{23}-\frac{14\!\cdots\!18}{21\!\cdots\!43}a^{22}+\frac{66\!\cdots\!35}{21\!\cdots\!43}a^{21}-\frac{21\!\cdots\!00}{21\!\cdots\!43}a^{20}+\frac{51\!\cdots\!75}{21\!\cdots\!43}a^{19}-\frac{94\!\cdots\!98}{21\!\cdots\!43}a^{18}+\frac{15\!\cdots\!33}{21\!\cdots\!43}a^{17}-\frac{25\!\cdots\!45}{21\!\cdots\!43}a^{16}+\frac{33\!\cdots\!51}{21\!\cdots\!43}a^{15}-\frac{45\!\cdots\!73}{21\!\cdots\!43}a^{14}+\frac{60\!\cdots\!56}{21\!\cdots\!43}a^{13}-\frac{69\!\cdots\!98}{21\!\cdots\!43}a^{12}+\frac{76\!\cdots\!21}{21\!\cdots\!43}a^{11}-\frac{86\!\cdots\!58}{21\!\cdots\!43}a^{10}+\frac{95\!\cdots\!24}{21\!\cdots\!43}a^{9}-\frac{10\!\cdots\!67}{21\!\cdots\!43}a^{8}+\frac{11\!\cdots\!66}{21\!\cdots\!43}a^{7}-\frac{10\!\cdots\!61}{21\!\cdots\!43}a^{6}+\frac{69\!\cdots\!22}{21\!\cdots\!43}a^{5}-\frac{35\!\cdots\!38}{21\!\cdots\!43}a^{4}+\frac{12\!\cdots\!19}{21\!\cdots\!43}a^{3}-\frac{25\!\cdots\!18}{21\!\cdots\!43}a^{2}+\frac{20\!\cdots\!02}{21\!\cdots\!43}a-\frac{81\!\cdots\!41}{21\!\cdots\!43}$, $\frac{23\!\cdots\!15}{43\!\cdots\!86}a^{23}-\frac{23\!\cdots\!81}{43\!\cdots\!86}a^{22}+\frac{11\!\cdots\!21}{43\!\cdots\!86}a^{21}-\frac{37\!\cdots\!97}{43\!\cdots\!86}a^{20}+\frac{45\!\cdots\!61}{21\!\cdots\!43}a^{19}-\frac{85\!\cdots\!64}{21\!\cdots\!43}a^{18}+\frac{28\!\cdots\!05}{43\!\cdots\!86}a^{17}-\frac{22\!\cdots\!41}{21\!\cdots\!43}a^{16}+\frac{31\!\cdots\!14}{21\!\cdots\!43}a^{15}-\frac{41\!\cdots\!00}{21\!\cdots\!43}a^{14}+\frac{55\!\cdots\!35}{21\!\cdots\!43}a^{13}-\frac{64\!\cdots\!06}{21\!\cdots\!43}a^{12}+\frac{14\!\cdots\!13}{43\!\cdots\!86}a^{11}-\frac{80\!\cdots\!83}{21\!\cdots\!43}a^{10}+\frac{88\!\cdots\!58}{21\!\cdots\!43}a^{9}-\frac{20\!\cdots\!81}{43\!\cdots\!86}a^{8}+\frac{11\!\cdots\!70}{21\!\cdots\!43}a^{7}-\frac{10\!\cdots\!84}{21\!\cdots\!43}a^{6}+\frac{13\!\cdots\!87}{43\!\cdots\!86}a^{5}-\frac{71\!\cdots\!27}{43\!\cdots\!86}a^{4}+\frac{13\!\cdots\!81}{21\!\cdots\!43}a^{3}-\frac{28\!\cdots\!79}{21\!\cdots\!43}a^{2}+\frac{16\!\cdots\!94}{21\!\cdots\!43}a-\frac{24\!\cdots\!07}{43\!\cdots\!86}$, $\frac{38\!\cdots\!19}{43\!\cdots\!86}a^{23}-\frac{39\!\cdots\!93}{43\!\cdots\!86}a^{22}+\frac{19\!\cdots\!25}{43\!\cdots\!86}a^{21}-\frac{64\!\cdots\!87}{43\!\cdots\!86}a^{20}+\frac{79\!\cdots\!00}{21\!\cdots\!43}a^{19}-\frac{15\!\cdots\!39}{21\!\cdots\!43}a^{18}+\frac{53\!\cdots\!91}{43\!\cdots\!86}a^{17}-\frac{42\!\cdots\!51}{21\!\cdots\!43}a^{16}+\frac{59\!\cdots\!06}{21\!\cdots\!43}a^{15}-\frac{80\!\cdots\!25}{21\!\cdots\!43}a^{14}+\frac{10\!\cdots\!36}{21\!\cdots\!43}a^{13}-\frac{12\!\cdots\!79}{21\!\cdots\!43}a^{12}+\frac{28\!\cdots\!27}{43\!\cdots\!86}a^{11}-\frac{16\!\cdots\!02}{21\!\cdots\!43}a^{10}+\frac{18\!\cdots\!49}{21\!\cdots\!43}a^{9}-\frac{40\!\cdots\!79}{43\!\cdots\!86}a^{8}+\frac{22\!\cdots\!30}{21\!\cdots\!43}a^{7}-\frac{21\!\cdots\!42}{21\!\cdots\!43}a^{6}+\frac{32\!\cdots\!61}{43\!\cdots\!86}a^{5}-\frac{19\!\cdots\!35}{43\!\cdots\!86}a^{4}+\frac{44\!\cdots\!48}{21\!\cdots\!43}a^{3}-\frac{14\!\cdots\!23}{21\!\cdots\!43}a^{2}+\frac{29\!\cdots\!23}{21\!\cdots\!43}a-\frac{76\!\cdots\!31}{43\!\cdots\!86}$, $\frac{67\!\cdots\!09}{43\!\cdots\!86}a^{23}-\frac{67\!\cdots\!69}{43\!\cdots\!86}a^{22}+\frac{32\!\cdots\!01}{43\!\cdots\!86}a^{21}-\frac{10\!\cdots\!47}{43\!\cdots\!86}a^{20}+\frac{12\!\cdots\!96}{21\!\cdots\!43}a^{19}-\frac{24\!\cdots\!57}{21\!\cdots\!43}a^{18}+\frac{81\!\cdots\!89}{43\!\cdots\!86}a^{17}-\frac{63\!\cdots\!50}{21\!\cdots\!43}a^{16}+\frac{87\!\cdots\!34}{21\!\cdots\!43}a^{15}-\frac{11\!\cdots\!39}{21\!\cdots\!43}a^{14}+\frac{15\!\cdots\!46}{21\!\cdots\!43}a^{13}-\frac{18\!\cdots\!30}{21\!\cdots\!43}a^{12}+\frac{39\!\cdots\!39}{43\!\cdots\!86}a^{11}-\frac{22\!\cdots\!73}{21\!\cdots\!43}a^{10}+\frac{24\!\cdots\!13}{21\!\cdots\!43}a^{9}-\frac{55\!\cdots\!43}{43\!\cdots\!86}a^{8}+\frac{30\!\cdots\!70}{21\!\cdots\!43}a^{7}-\frac{27\!\cdots\!96}{21\!\cdots\!43}a^{6}+\frac{37\!\cdots\!83}{43\!\cdots\!86}a^{5}-\frac{18\!\cdots\!45}{43\!\cdots\!86}a^{4}+\frac{30\!\cdots\!38}{21\!\cdots\!43}a^{3}-\frac{53\!\cdots\!14}{21\!\cdots\!43}a^{2}-\frac{19\!\cdots\!84}{21\!\cdots\!43}a-\frac{46\!\cdots\!13}{43\!\cdots\!86}$, $\frac{14\!\cdots\!63}{43\!\cdots\!86}a^{23}-\frac{14\!\cdots\!43}{43\!\cdots\!86}a^{22}+\frac{68\!\cdots\!33}{43\!\cdots\!86}a^{21}-\frac{22\!\cdots\!63}{43\!\cdots\!86}a^{20}+\frac{27\!\cdots\!09}{21\!\cdots\!43}a^{19}-\frac{50\!\cdots\!34}{21\!\cdots\!43}a^{18}+\frac{17\!\cdots\!47}{43\!\cdots\!86}a^{17}-\frac{13\!\cdots\!46}{21\!\cdots\!43}a^{16}+\frac{18\!\cdots\!31}{21\!\cdots\!43}a^{15}-\frac{24\!\cdots\!85}{21\!\cdots\!43}a^{14}+\frac{32\!\cdots\!40}{21\!\cdots\!43}a^{13}-\frac{38\!\cdots\!40}{21\!\cdots\!43}a^{12}+\frac{83\!\cdots\!15}{43\!\cdots\!86}a^{11}-\frac{47\!\cdots\!01}{21\!\cdots\!43}a^{10}+\frac{52\!\cdots\!72}{21\!\cdots\!43}a^{9}-\frac{11\!\cdots\!87}{43\!\cdots\!86}a^{8}+\frac{64\!\cdots\!34}{21\!\cdots\!43}a^{7}-\frac{58\!\cdots\!59}{21\!\cdots\!43}a^{6}+\frac{79\!\cdots\!19}{43\!\cdots\!86}a^{5}-\frac{40\!\cdots\!91}{43\!\cdots\!86}a^{4}+\frac{74\!\cdots\!20}{21\!\cdots\!43}a^{3}-\frac{17\!\cdots\!50}{21\!\cdots\!43}a^{2}+\frac{19\!\cdots\!65}{21\!\cdots\!43}a-\frac{50\!\cdots\!09}{43\!\cdots\!86}$, $\frac{47\!\cdots\!41}{87\!\cdots\!72}a^{23}-\frac{47\!\cdots\!03}{87\!\cdots\!72}a^{22}+\frac{22\!\cdots\!23}{87\!\cdots\!72}a^{21}-\frac{75\!\cdots\!33}{87\!\cdots\!72}a^{20}+\frac{90\!\cdots\!87}{43\!\cdots\!86}a^{19}-\frac{85\!\cdots\!77}{21\!\cdots\!43}a^{18}+\frac{58\!\cdots\!53}{87\!\cdots\!72}a^{17}-\frac{22\!\cdots\!77}{21\!\cdots\!43}a^{16}+\frac{62\!\cdots\!97}{43\!\cdots\!86}a^{15}-\frac{84\!\cdots\!15}{43\!\cdots\!86}a^{14}+\frac{56\!\cdots\!72}{21\!\cdots\!43}a^{13}-\frac{13\!\cdots\!13}{43\!\cdots\!86}a^{12}+\frac{28\!\cdots\!47}{87\!\cdots\!72}a^{11}-\frac{81\!\cdots\!36}{21\!\cdots\!43}a^{10}+\frac{18\!\cdots\!53}{43\!\cdots\!86}a^{9}-\frac{40\!\cdots\!73}{87\!\cdots\!72}a^{8}+\frac{22\!\cdots\!47}{43\!\cdots\!86}a^{7}-\frac{10\!\cdots\!07}{21\!\cdots\!43}a^{6}+\frac{28\!\cdots\!71}{87\!\cdots\!72}a^{5}-\frac{15\!\cdots\!77}{87\!\cdots\!72}a^{4}+\frac{14\!\cdots\!12}{21\!\cdots\!43}a^{3}-\frac{33\!\cdots\!11}{21\!\cdots\!43}a^{2}+\frac{25\!\cdots\!79}{21\!\cdots\!43}a-\frac{34\!\cdots\!07}{87\!\cdots\!72}$, $\frac{28\!\cdots\!43}{43\!\cdots\!86}a^{23}-\frac{22\!\cdots\!11}{43\!\cdots\!86}a^{22}+\frac{81\!\cdots\!71}{43\!\cdots\!86}a^{21}-\frac{19\!\cdots\!93}{43\!\cdots\!86}a^{20}+\frac{12\!\cdots\!61}{21\!\cdots\!43}a^{19}-\frac{49\!\cdots\!40}{21\!\cdots\!43}a^{18}-\frac{42\!\cdots\!01}{43\!\cdots\!86}a^{17}+\frac{21\!\cdots\!43}{21\!\cdots\!43}a^{16}-\frac{81\!\cdots\!50}{21\!\cdots\!43}a^{15}+\frac{10\!\cdots\!55}{21\!\cdots\!43}a^{14}-\frac{14\!\cdots\!25}{21\!\cdots\!43}a^{13}+\frac{30\!\cdots\!17}{21\!\cdots\!43}a^{12}-\frac{72\!\cdots\!77}{43\!\cdots\!86}a^{11}+\frac{35\!\cdots\!53}{21\!\cdots\!43}a^{10}-\frac{43\!\cdots\!43}{21\!\cdots\!43}a^{9}+\frac{89\!\cdots\!13}{43\!\cdots\!86}a^{8}-\frac{55\!\cdots\!70}{21\!\cdots\!43}a^{7}+\frac{84\!\cdots\!17}{21\!\cdots\!43}a^{6}-\frac{18\!\cdots\!43}{43\!\cdots\!86}a^{5}+\frac{12\!\cdots\!17}{43\!\cdots\!86}a^{4}-\frac{31\!\cdots\!11}{21\!\cdots\!43}a^{3}+\frac{10\!\cdots\!61}{21\!\cdots\!43}a^{2}-\frac{10\!\cdots\!75}{21\!\cdots\!43}a-\frac{93\!\cdots\!09}{43\!\cdots\!86}$, $\frac{20\!\cdots\!85}{87\!\cdots\!72}a^{23}-\frac{14\!\cdots\!87}{87\!\cdots\!72}a^{22}+\frac{47\!\cdots\!31}{87\!\cdots\!72}a^{21}-\frac{98\!\cdots\!21}{87\!\cdots\!72}a^{20}+\frac{43\!\cdots\!85}{43\!\cdots\!86}a^{19}+\frac{21\!\cdots\!01}{21\!\cdots\!43}a^{18}-\frac{23\!\cdots\!87}{87\!\cdots\!72}a^{17}+\frac{15\!\cdots\!04}{21\!\cdots\!43}a^{16}-\frac{75\!\cdots\!39}{43\!\cdots\!86}a^{15}+\frac{85\!\cdots\!29}{43\!\cdots\!86}a^{14}-\frac{65\!\cdots\!46}{21\!\cdots\!43}a^{13}+\frac{23\!\cdots\!57}{43\!\cdots\!86}a^{12}-\frac{49\!\cdots\!61}{87\!\cdots\!72}a^{11}+\frac{13\!\cdots\!98}{21\!\cdots\!43}a^{10}-\frac{30\!\cdots\!51}{43\!\cdots\!86}a^{9}+\frac{61\!\cdots\!23}{87\!\cdots\!72}a^{8}-\frac{40\!\cdots\!81}{43\!\cdots\!86}a^{7}+\frac{28\!\cdots\!92}{21\!\cdots\!43}a^{6}-\frac{11\!\cdots\!53}{87\!\cdots\!72}a^{5}+\frac{68\!\cdots\!83}{87\!\cdots\!72}a^{4}-\frac{63\!\cdots\!78}{21\!\cdots\!43}a^{3}+\frac{67\!\cdots\!34}{21\!\cdots\!43}a^{2}+\frac{60\!\cdots\!46}{21\!\cdots\!43}a-\frac{98\!\cdots\!59}{87\!\cdots\!72}$, $\frac{41\!\cdots\!45}{87\!\cdots\!72}a^{23}-\frac{41\!\cdots\!15}{87\!\cdots\!72}a^{22}+\frac{19\!\cdots\!19}{87\!\cdots\!72}a^{21}-\frac{62\!\cdots\!33}{87\!\cdots\!72}a^{20}+\frac{74\!\cdots\!77}{43\!\cdots\!86}a^{19}-\frac{69\!\cdots\!38}{21\!\cdots\!43}a^{18}+\frac{46\!\cdots\!41}{87\!\cdots\!72}a^{17}-\frac{18\!\cdots\!09}{21\!\cdots\!43}a^{16}+\frac{49\!\cdots\!31}{43\!\cdots\!86}a^{15}-\frac{66\!\cdots\!01}{43\!\cdots\!86}a^{14}+\frac{44\!\cdots\!10}{21\!\cdots\!43}a^{13}-\frac{10\!\cdots\!03}{43\!\cdots\!86}a^{12}+\frac{22\!\cdots\!87}{87\!\cdots\!72}a^{11}-\frac{63\!\cdots\!95}{21\!\cdots\!43}a^{10}+\frac{13\!\cdots\!13}{43\!\cdots\!86}a^{9}-\frac{31\!\cdots\!09}{87\!\cdots\!72}a^{8}+\frac{17\!\cdots\!89}{43\!\cdots\!86}a^{7}-\frac{77\!\cdots\!39}{21\!\cdots\!43}a^{6}+\frac{20\!\cdots\!79}{87\!\cdots\!72}a^{5}-\frac{10\!\cdots\!81}{87\!\cdots\!72}a^{4}+\frac{97\!\cdots\!21}{21\!\cdots\!43}a^{3}-\frac{22\!\cdots\!59}{21\!\cdots\!43}a^{2}+\frac{29\!\cdots\!49}{21\!\cdots\!43}a-\frac{29\!\cdots\!75}{87\!\cdots\!72}$, $\frac{27\!\cdots\!59}{87\!\cdots\!72}a^{23}-\frac{26\!\cdots\!13}{87\!\cdots\!72}a^{22}+\frac{12\!\cdots\!93}{87\!\cdots\!72}a^{21}-\frac{40\!\cdots\!51}{87\!\cdots\!72}a^{20}+\frac{47\!\cdots\!37}{43\!\cdots\!86}a^{19}-\frac{43\!\cdots\!05}{21\!\cdots\!43}a^{18}+\frac{29\!\cdots\!39}{87\!\cdots\!72}a^{17}-\frac{11\!\cdots\!46}{21\!\cdots\!43}a^{16}+\frac{31\!\cdots\!65}{43\!\cdots\!86}a^{15}-\frac{42\!\cdots\!31}{43\!\cdots\!86}a^{14}+\frac{27\!\cdots\!65}{21\!\cdots\!43}a^{13}-\frac{64\!\cdots\!21}{43\!\cdots\!86}a^{12}+\frac{14\!\cdots\!53}{87\!\cdots\!72}a^{11}-\frac{39\!\cdots\!07}{21\!\cdots\!43}a^{10}+\frac{87\!\cdots\!83}{43\!\cdots\!86}a^{9}-\frac{19\!\cdots\!43}{87\!\cdots\!72}a^{8}+\frac{10\!\cdots\!19}{43\!\cdots\!86}a^{7}-\frac{47\!\cdots\!35}{21\!\cdots\!43}a^{6}+\frac{12\!\cdots\!73}{87\!\cdots\!72}a^{5}-\frac{63\!\cdots\!15}{87\!\cdots\!72}a^{4}+\frac{54\!\cdots\!04}{21\!\cdots\!43}a^{3}-\frac{97\!\cdots\!07}{21\!\cdots\!43}a^{2}+\frac{13\!\cdots\!02}{21\!\cdots\!43}a+\frac{60\!\cdots\!39}{87\!\cdots\!72}$, $\frac{65\!\cdots\!87}{43\!\cdots\!86}a^{23}-\frac{66\!\cdots\!99}{43\!\cdots\!86}a^{22}+\frac{32\!\cdots\!85}{43\!\cdots\!86}a^{21}-\frac{10\!\cdots\!45}{43\!\cdots\!86}a^{20}+\frac{13\!\cdots\!89}{21\!\cdots\!43}a^{19}-\frac{24\!\cdots\!49}{21\!\cdots\!43}a^{18}+\frac{83\!\cdots\!39}{43\!\cdots\!86}a^{17}-\frac{65\!\cdots\!17}{21\!\cdots\!43}a^{16}+\frac{90\!\cdots\!60}{21\!\cdots\!43}a^{15}-\frac{12\!\cdots\!26}{21\!\cdots\!43}a^{14}+\frac{16\!\cdots\!89}{21\!\cdots\!43}a^{13}-\frac{18\!\cdots\!67}{21\!\cdots\!43}a^{12}+\frac{41\!\cdots\!91}{43\!\cdots\!86}a^{11}-\frac{23\!\cdots\!66}{21\!\cdots\!43}a^{10}+\frac{25\!\cdots\!99}{21\!\cdots\!43}a^{9}-\frac{58\!\cdots\!39}{43\!\cdots\!86}a^{8}+\frac{32\!\cdots\!34}{21\!\cdots\!43}a^{7}-\frac{29\!\cdots\!66}{21\!\cdots\!43}a^{6}+\frac{39\!\cdots\!69}{43\!\cdots\!86}a^{5}-\frac{19\!\cdots\!47}{43\!\cdots\!86}a^{4}+\frac{34\!\cdots\!42}{21\!\cdots\!43}a^{3}-\frac{63\!\cdots\!65}{21\!\cdots\!43}a^{2}+\frac{65\!\cdots\!23}{21\!\cdots\!43}a-\frac{48\!\cdots\!77}{43\!\cdots\!86}$
| 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: | \( 8548806.54922059 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 8548806.54922059 \cdot 1}{2\cdot\sqrt{940350029043078386535797119140625}}\cr\approx \mathstrut & 0.213869763394454 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*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 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*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 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*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 - 10*x^23 + 48*x^22 - 158*x^21 + 381*x^20 - 714*x^19 + 1213*x^18 - 1907*x^17 + 2614*x^16 - 3516*x^15 + 4694*x^14 - 5466*x^13 + 6009*x^12 - 6785*x^11 + 7506*x^10 - 8491*x^9 + 9365*x^8 - 8498*x^7 + 5899*x^6 - 3118*x^5 + 1203*x^4 - 296*x^3 + 36*x^2 - 7*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
$\SL(2,5):C_2$ (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
| 6.2.14250625.2, 12.4.203080312890625.1 |
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: | 24.4.940350029043078386535797119140625.2 |
| 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | R | $20{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }^{6}$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.3.0.1}{3} }^{8}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.3.0.1}{3} }^{8}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.8.1 | $x^{12} + 12 x^{10} + 32 x^{9} + 54 x^{8} + 96 x^{7} - 50 x^{6} + 240 x^{5} - 360 x^{4} - 884 x^{3} + 4044 x^{2} - 3912 x + 4173$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
| 5.12.8.1 | $x^{12} + 12 x^{10} + 32 x^{9} + 54 x^{8} + 96 x^{7} - 50 x^{6} + 240 x^{5} - 360 x^{4} - 884 x^{3} + 4044 x^{2} - 3912 x + 4173$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(151\)
| 151.4.2.2 | $x^{4} - 235711 x^{3} - 1406251373 x^{2} - 129486879 x + 136806$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 151.4.0.1 | $x^{4} + 13 x^{2} + 89 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 151.8.4.1 | $x^{8} + 630 x^{6} + 178 x^{5} + 140913 x^{4} - 51442 x^{3} + 13227221 x^{2} - 11825252 x + 435668407$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 151.8.4.1 | $x^{8} + 630 x^{6} + 178 x^{5} + 140913 x^{4} - 51442 x^{3} + 13227221 x^{2} - 11825252 x + 435668407$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.151.2t1.a.a | $1$ | $ 151 $ | \(\Q(\sqrt{-151}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.3775.120.a.a | $2$ | $ 5^{2} \cdot 151 $ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3775.120.a.b | $2$ | $ 5^{2} \cdot 151 $ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3775.120.a.c | $2$ | $ 5^{2} \cdot 151 $ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3775.120.a.d | $2$ | $ 5^{2} \cdot 151 $ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| * | 3.3775.12t76.a.a | $3$ | $ 5^{2} \cdot 151 $ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| * | 3.3775.12t76.a.b | $3$ | $ 5^{2} \cdot 151 $ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| 3.570025.12t33.a.a | $3$ | $ 5^{2} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 3.570025.12t33.a.b | $3$ | $ 5^{2} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 4.570025.10t11.a.a | $4$ | $ 5^{2} \cdot 151^{2}$ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
| 4.570025.5t4.a.a | $4$ | $ 5^{2} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
| 4.570025.40t188.a.a | $4$ | $ 5^{2} \cdot 151^{2}$ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 4.570025.40t188.a.b | $4$ | $ 5^{2} \cdot 151^{2}$ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 5.2151844375.12t75.a.a | $5$ | $ 5^{4} \cdot 151^{3}$ | 10.0.49064203594375.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
| * | 5.14250625.6t12.a.a | $5$ | $ 5^{4} \cdot 151^{2}$ | 5.1.570025.1 | $A_5$ (as 5T4) | $1$ | $1$ |
| * | 6.2151844375.24t576.a.a | $6$ | $ 5^{4} \cdot 151^{3}$ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
| * | 6.2151844375.24t576.a.b | $6$ | $ 5^{4} \cdot 151^{3}$ | 24.4.940350029043078386535797119140625.1 | $\SL(2,5):C_2$ (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 *.