Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 12*x^23 + 74*x^22 - 308*x^21 + 966*x^20 - 2412*x^19 + 4884*x^18 - 7968*x^17 + 10016*x^16 - 8392*x^15 + 820*x^14 + 12680*x^13 - 29520*x^12 + 44840*x^11 - 53520*x^10 + 50544*x^9 - 38528*x^8 + 24016*x^7 - 13652*x^6 + 7544*x^5 - 3696*x^4 + 1472*x^3 - 384*x^2 + 88*x + 4)
gp: K = bnfinit(y^24 - 12*y^23 + 74*y^22 - 308*y^21 + 966*y^20 - 2412*y^19 + 4884*y^18 - 7968*y^17 + 10016*y^16 - 8392*y^15 + 820*y^14 + 12680*y^13 - 29520*y^12 + 44840*y^11 - 53520*y^10 + 50544*y^9 - 38528*y^8 + 24016*y^7 - 13652*y^6 + 7544*y^5 - 3696*y^4 + 1472*y^3 - 384*y^2 + 88*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 12*x^23 + 74*x^22 - 308*x^21 + 966*x^20 - 2412*x^19 + 4884*x^18 - 7968*x^17 + 10016*x^16 - 8392*x^15 + 820*x^14 + 12680*x^13 - 29520*x^12 + 44840*x^11 - 53520*x^10 + 50544*x^9 - 38528*x^8 + 24016*x^7 - 13652*x^6 + 7544*x^5 - 3696*x^4 + 1472*x^3 - 384*x^2 + 88*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 12*x^23 + 74*x^22 - 308*x^21 + 966*x^20 - 2412*x^19 + 4884*x^18 - 7968*x^17 + 10016*x^16 - 8392*x^15 + 820*x^14 + 12680*x^13 - 29520*x^12 + 44840*x^11 - 53520*x^10 + 50544*x^9 - 38528*x^8 + 24016*x^7 - 13652*x^6 + 7544*x^5 - 3696*x^4 + 1472*x^3 - 384*x^2 + 88*x + 4)
\( x^{24} - 12 x^{23} + 74 x^{22} - 308 x^{21} + 966 x^{20} - 2412 x^{19} + 4884 x^{18} - 7968 x^{17} + \cdots + 4 \)
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: |
\(1280000000000000000000000000000000\)
\(\medspace = 2^{38}\cdot 5^{31}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.96\) | 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: | $2^{19/12}5^{31/20}\approx 36.31067590725307$ | ||
| Ramified primes: |
\(2\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{4}a^{19}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{20}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}$, $\frac{1}{4}a^{21}-\frac{1}{2}a^{9}$, $\frac{1}{4}a^{22}-\frac{1}{2}a^{10}$, $\frac{1}{63\!\cdots\!56}a^{23}-\frac{17\!\cdots\!91}{31\!\cdots\!78}a^{22}-\frac{64\!\cdots\!57}{63\!\cdots\!56}a^{21}-\frac{58\!\cdots\!70}{15\!\cdots\!89}a^{20}+\frac{17\!\cdots\!93}{31\!\cdots\!78}a^{19}+\frac{21\!\cdots\!14}{15\!\cdots\!89}a^{18}+\frac{51\!\cdots\!95}{15\!\cdots\!89}a^{17}+\frac{25\!\cdots\!25}{31\!\cdots\!78}a^{16}-\frac{72\!\cdots\!88}{12\!\cdots\!53}a^{15}-\frac{13\!\cdots\!34}{15\!\cdots\!89}a^{14}+\frac{60\!\cdots\!97}{31\!\cdots\!78}a^{13}-\frac{99\!\cdots\!98}{15\!\cdots\!89}a^{12}-\frac{23\!\cdots\!35}{31\!\cdots\!78}a^{11}+\frac{70\!\cdots\!81}{15\!\cdots\!89}a^{10}+\frac{86\!\cdots\!09}{31\!\cdots\!78}a^{9}+\frac{31\!\cdots\!76}{15\!\cdots\!89}a^{8}-\frac{48\!\cdots\!39}{15\!\cdots\!89}a^{7}+\frac{68\!\cdots\!18}{15\!\cdots\!89}a^{6}+\frac{29\!\cdots\!08}{12\!\cdots\!53}a^{5}-\frac{14\!\cdots\!45}{15\!\cdots\!89}a^{4}+\frac{57\!\cdots\!71}{15\!\cdots\!89}a^{3}+\frac{39\!\cdots\!14}{15\!\cdots\!89}a^{2}+\frac{41\!\cdots\!03}{15\!\cdots\!89}a-\frac{50\!\cdots\!23}{15\!\cdots\!89}$
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: |
$\frac{11\!\cdots\!26}{12\!\cdots\!53}a^{23}+\frac{14\!\cdots\!74}{12\!\cdots\!53}a^{22}-\frac{18\!\cdots\!47}{24\!\cdots\!06}a^{21}+\frac{40\!\cdots\!46}{12\!\cdots\!53}a^{20}-\frac{13\!\cdots\!19}{12\!\cdots\!53}a^{19}+\frac{34\!\cdots\!46}{12\!\cdots\!53}a^{18}-\frac{71\!\cdots\!32}{12\!\cdots\!53}a^{17}+\frac{12\!\cdots\!55}{12\!\cdots\!53}a^{16}-\frac{15\!\cdots\!23}{12\!\cdots\!53}a^{15}+\frac{14\!\cdots\!69}{12\!\cdots\!53}a^{14}-\frac{38\!\cdots\!65}{12\!\cdots\!53}a^{13}-\frac{16\!\cdots\!05}{12\!\cdots\!53}a^{12}+\frac{44\!\cdots\!40}{12\!\cdots\!53}a^{11}-\frac{69\!\cdots\!69}{12\!\cdots\!53}a^{10}+\frac{85\!\cdots\!54}{12\!\cdots\!53}a^{9}-\frac{83\!\cdots\!17}{12\!\cdots\!53}a^{8}+\frac{64\!\cdots\!72}{12\!\cdots\!53}a^{7}-\frac{38\!\cdots\!35}{12\!\cdots\!53}a^{6}+\frac{20\!\cdots\!08}{12\!\cdots\!53}a^{5}-\frac{10\!\cdots\!85}{12\!\cdots\!53}a^{4}+\frac{52\!\cdots\!38}{12\!\cdots\!53}a^{3}-\frac{17\!\cdots\!79}{12\!\cdots\!53}a^{2}+\frac{20\!\cdots\!28}{12\!\cdots\!53}a-\frac{11\!\cdots\!17}{12\!\cdots\!53}$, $\frac{32\!\cdots\!72}{12\!\cdots\!53}a^{23}+\frac{38\!\cdots\!91}{12\!\cdots\!53}a^{22}-\frac{46\!\cdots\!59}{24\!\cdots\!06}a^{21}+\frac{19\!\cdots\!85}{24\!\cdots\!06}a^{20}-\frac{30\!\cdots\!27}{12\!\cdots\!53}a^{19}+\frac{74\!\cdots\!68}{12\!\cdots\!53}a^{18}-\frac{15\!\cdots\!20}{12\!\cdots\!53}a^{17}+\frac{24\!\cdots\!84}{12\!\cdots\!53}a^{16}-\frac{30\!\cdots\!09}{12\!\cdots\!53}a^{15}+\frac{24\!\cdots\!32}{12\!\cdots\!53}a^{14}-\frac{62\!\cdots\!55}{12\!\cdots\!53}a^{13}-\frac{40\!\cdots\!88}{12\!\cdots\!53}a^{12}+\frac{91\!\cdots\!48}{12\!\cdots\!53}a^{11}-\frac{13\!\cdots\!07}{12\!\cdots\!53}a^{10}+\frac{16\!\cdots\!52}{12\!\cdots\!53}a^{9}-\frac{14\!\cdots\!41}{12\!\cdots\!53}a^{8}+\frac{11\!\cdots\!60}{12\!\cdots\!53}a^{7}-\frac{69\!\cdots\!13}{12\!\cdots\!53}a^{6}+\frac{39\!\cdots\!40}{12\!\cdots\!53}a^{5}-\frac{21\!\cdots\!23}{12\!\cdots\!53}a^{4}+\frac{10\!\cdots\!90}{12\!\cdots\!53}a^{3}-\frac{42\!\cdots\!71}{12\!\cdots\!53}a^{2}+\frac{12\!\cdots\!60}{12\!\cdots\!53}a-\frac{23\!\cdots\!71}{12\!\cdots\!53}$, $\frac{39\!\cdots\!85}{31\!\cdots\!78}a^{23}+\frac{24\!\cdots\!88}{15\!\cdots\!89}a^{22}-\frac{15\!\cdots\!76}{15\!\cdots\!89}a^{21}+\frac{25\!\cdots\!09}{63\!\cdots\!56}a^{20}-\frac{20\!\cdots\!23}{15\!\cdots\!89}a^{19}+\frac{49\!\cdots\!37}{15\!\cdots\!89}a^{18}-\frac{19\!\cdots\!17}{31\!\cdots\!78}a^{17}+\frac{30\!\cdots\!83}{31\!\cdots\!78}a^{16}-\frac{13\!\cdots\!88}{12\!\cdots\!53}a^{15}+\frac{24\!\cdots\!47}{31\!\cdots\!78}a^{14}+\frac{56\!\cdots\!07}{15\!\cdots\!89}a^{13}-\frac{66\!\cdots\!53}{31\!\cdots\!78}a^{12}+\frac{12\!\cdots\!67}{31\!\cdots\!78}a^{11}-\frac{85\!\cdots\!94}{15\!\cdots\!89}a^{10}+\frac{91\!\cdots\!06}{15\!\cdots\!89}a^{9}-\frac{15\!\cdots\!15}{31\!\cdots\!78}a^{8}+\frac{44\!\cdots\!87}{15\!\cdots\!89}a^{7}-\frac{19\!\cdots\!34}{15\!\cdots\!89}a^{6}+\frac{67\!\cdots\!71}{12\!\cdots\!53}a^{5}-\frac{67\!\cdots\!73}{15\!\cdots\!89}a^{4}+\frac{25\!\cdots\!00}{15\!\cdots\!89}a^{3}+\frac{39\!\cdots\!12}{15\!\cdots\!89}a^{2}-\frac{49\!\cdots\!25}{15\!\cdots\!89}a-\frac{10\!\cdots\!15}{15\!\cdots\!89}$, $\frac{13\!\cdots\!49}{63\!\cdots\!56}a^{23}-\frac{36\!\cdots\!65}{15\!\cdots\!89}a^{22}+\frac{41\!\cdots\!19}{31\!\cdots\!78}a^{21}-\frac{31\!\cdots\!97}{63\!\cdots\!56}a^{20}+\frac{45\!\cdots\!45}{31\!\cdots\!78}a^{19}-\frac{10\!\cdots\!39}{31\!\cdots\!78}a^{18}+\frac{19\!\cdots\!33}{31\!\cdots\!78}a^{17}-\frac{27\!\cdots\!95}{31\!\cdots\!78}a^{16}+\frac{21\!\cdots\!01}{24\!\cdots\!06}a^{15}-\frac{12\!\cdots\!93}{31\!\cdots\!78}a^{14}-\frac{21\!\cdots\!91}{31\!\cdots\!78}a^{13}+\frac{66\!\cdots\!55}{31\!\cdots\!78}a^{12}-\frac{56\!\cdots\!70}{15\!\cdots\!89}a^{11}+\frac{71\!\cdots\!97}{15\!\cdots\!89}a^{10}-\frac{73\!\cdots\!46}{15\!\cdots\!89}a^{9}+\frac{11\!\cdots\!97}{31\!\cdots\!78}a^{8}-\frac{35\!\cdots\!08}{15\!\cdots\!89}a^{7}+\frac{19\!\cdots\!34}{15\!\cdots\!89}a^{6}-\frac{11\!\cdots\!79}{12\!\cdots\!53}a^{5}+\frac{85\!\cdots\!14}{15\!\cdots\!89}a^{4}-\frac{32\!\cdots\!09}{15\!\cdots\!89}a^{3}+\frac{10\!\cdots\!73}{15\!\cdots\!89}a^{2}-\frac{46\!\cdots\!74}{15\!\cdots\!89}a+\frac{30\!\cdots\!45}{15\!\cdots\!89}$, $\frac{87\!\cdots\!43}{63\!\cdots\!56}a^{23}-\frac{10\!\cdots\!01}{63\!\cdots\!56}a^{22}+\frac{16\!\cdots\!74}{15\!\cdots\!89}a^{21}-\frac{27\!\cdots\!85}{63\!\cdots\!56}a^{20}+\frac{22\!\cdots\!34}{15\!\cdots\!89}a^{19}-\frac{56\!\cdots\!15}{15\!\cdots\!89}a^{18}+\frac{23\!\cdots\!39}{31\!\cdots\!78}a^{17}-\frac{38\!\cdots\!73}{31\!\cdots\!78}a^{16}+\frac{18\!\cdots\!36}{12\!\cdots\!53}a^{15}-\frac{21\!\cdots\!86}{15\!\cdots\!89}a^{14}+\frac{75\!\cdots\!93}{31\!\cdots\!78}a^{13}+\frac{57\!\cdots\!27}{31\!\cdots\!78}a^{12}-\frac{70\!\cdots\!68}{15\!\cdots\!89}a^{11}+\frac{21\!\cdots\!35}{31\!\cdots\!78}a^{10}-\frac{13\!\cdots\!31}{15\!\cdots\!89}a^{9}+\frac{25\!\cdots\!19}{31\!\cdots\!78}a^{8}-\frac{96\!\cdots\!64}{15\!\cdots\!89}a^{7}+\frac{58\!\cdots\!16}{15\!\cdots\!89}a^{6}-\frac{24\!\cdots\!83}{12\!\cdots\!53}a^{5}+\frac{16\!\cdots\!15}{15\!\cdots\!89}a^{4}-\frac{82\!\cdots\!51}{15\!\cdots\!89}a^{3}+\frac{29\!\cdots\!30}{15\!\cdots\!89}a^{2}-\frac{68\!\cdots\!73}{15\!\cdots\!89}a+\frac{30\!\cdots\!61}{15\!\cdots\!89}$, $\frac{28\!\cdots\!05}{24\!\cdots\!06}a^{23}-\frac{64\!\cdots\!31}{49\!\cdots\!12}a^{22}+\frac{93\!\cdots\!27}{12\!\cdots\!53}a^{21}-\frac{72\!\cdots\!55}{24\!\cdots\!06}a^{20}+\frac{10\!\cdots\!00}{12\!\cdots\!53}a^{19}-\frac{49\!\cdots\!73}{24\!\cdots\!06}a^{18}+\frac{92\!\cdots\!97}{24\!\cdots\!06}a^{17}-\frac{67\!\cdots\!39}{12\!\cdots\!53}a^{16}+\frac{68\!\cdots\!38}{12\!\cdots\!53}a^{15}-\frac{29\!\cdots\!68}{12\!\cdots\!53}a^{14}-\frac{58\!\cdots\!27}{12\!\cdots\!53}a^{13}+\frac{35\!\cdots\!67}{24\!\cdots\!06}a^{12}-\frac{28\!\cdots\!45}{12\!\cdots\!53}a^{11}+\frac{71\!\cdots\!21}{24\!\cdots\!06}a^{10}-\frac{34\!\cdots\!64}{12\!\cdots\!53}a^{9}+\frac{24\!\cdots\!18}{12\!\cdots\!53}a^{8}-\frac{11\!\cdots\!10}{12\!\cdots\!53}a^{7}+\frac{31\!\cdots\!70}{12\!\cdots\!53}a^{6}-\frac{13\!\cdots\!09}{12\!\cdots\!53}a^{5}+\frac{59\!\cdots\!02}{12\!\cdots\!53}a^{4}+\frac{37\!\cdots\!06}{12\!\cdots\!53}a^{3}-\frac{72\!\cdots\!40}{12\!\cdots\!53}a^{2}+\frac{35\!\cdots\!72}{12\!\cdots\!53}a-\frac{46\!\cdots\!06}{12\!\cdots\!53}$, $\frac{14\!\cdots\!75}{31\!\cdots\!78}a^{23}-\frac{85\!\cdots\!79}{15\!\cdots\!89}a^{22}+\frac{20\!\cdots\!45}{63\!\cdots\!56}a^{21}-\frac{83\!\cdots\!83}{63\!\cdots\!56}a^{20}+\frac{25\!\cdots\!33}{63\!\cdots\!56}a^{19}-\frac{30\!\cdots\!47}{31\!\cdots\!78}a^{18}+\frac{29\!\cdots\!48}{15\!\cdots\!89}a^{17}-\frac{46\!\cdots\!59}{15\!\cdots\!89}a^{16}+\frac{82\!\cdots\!47}{24\!\cdots\!06}a^{15}-\frac{71\!\cdots\!15}{31\!\cdots\!78}a^{14}-\frac{16\!\cdots\!99}{15\!\cdots\!89}a^{13}+\frac{19\!\cdots\!75}{31\!\cdots\!78}a^{12}-\frac{18\!\cdots\!10}{15\!\cdots\!89}a^{11}+\frac{50\!\cdots\!25}{31\!\cdots\!78}a^{10}-\frac{55\!\cdots\!61}{31\!\cdots\!78}a^{9}+\frac{46\!\cdots\!77}{31\!\cdots\!78}a^{8}-\frac{30\!\cdots\!81}{31\!\cdots\!78}a^{7}+\frac{77\!\cdots\!61}{15\!\cdots\!89}a^{6}-\frac{30\!\cdots\!16}{12\!\cdots\!53}a^{5}+\frac{22\!\cdots\!44}{15\!\cdots\!89}a^{4}-\frac{87\!\cdots\!02}{15\!\cdots\!89}a^{3}+\frac{17\!\cdots\!48}{15\!\cdots\!89}a^{2}+\frac{22\!\cdots\!57}{15\!\cdots\!89}a-\frac{50\!\cdots\!13}{15\!\cdots\!89}$, $\frac{13\!\cdots\!19}{31\!\cdots\!78}a^{23}-\frac{31\!\cdots\!11}{63\!\cdots\!56}a^{22}+\frac{48\!\cdots\!35}{15\!\cdots\!89}a^{21}-\frac{38\!\cdots\!19}{31\!\cdots\!78}a^{20}+\frac{23\!\cdots\!91}{63\!\cdots\!56}a^{19}-\frac{26\!\cdots\!87}{31\!\cdots\!78}a^{18}+\frac{49\!\cdots\!69}{31\!\cdots\!78}a^{17}-\frac{34\!\cdots\!12}{15\!\cdots\!89}a^{16}+\frac{46\!\cdots\!75}{24\!\cdots\!06}a^{15}-\frac{71\!\cdots\!13}{31\!\cdots\!78}a^{14}-\frac{11\!\cdots\!55}{31\!\cdots\!78}a^{13}+\frac{24\!\cdots\!63}{31\!\cdots\!78}a^{12}-\frac{34\!\cdots\!81}{31\!\cdots\!78}a^{11}+\frac{17\!\cdots\!04}{15\!\cdots\!89}a^{10}-\frac{13\!\cdots\!22}{15\!\cdots\!89}a^{9}+\frac{32\!\cdots\!93}{15\!\cdots\!89}a^{8}+\frac{14\!\cdots\!51}{31\!\cdots\!78}a^{7}-\frac{10\!\cdots\!51}{15\!\cdots\!89}a^{6}+\frac{59\!\cdots\!83}{12\!\cdots\!53}a^{5}-\frac{30\!\cdots\!53}{15\!\cdots\!89}a^{4}+\frac{16\!\cdots\!97}{15\!\cdots\!89}a^{3}-\frac{96\!\cdots\!45}{15\!\cdots\!89}a^{2}+\frac{25\!\cdots\!06}{15\!\cdots\!89}a-\frac{41\!\cdots\!69}{15\!\cdots\!89}$, $\frac{88\!\cdots\!69}{31\!\cdots\!78}a^{23}-\frac{21\!\cdots\!87}{63\!\cdots\!56}a^{22}+\frac{13\!\cdots\!53}{63\!\cdots\!56}a^{21}-\frac{55\!\cdots\!19}{63\!\cdots\!56}a^{20}+\frac{17\!\cdots\!21}{63\!\cdots\!56}a^{19}-\frac{10\!\cdots\!64}{15\!\cdots\!89}a^{18}+\frac{41\!\cdots\!45}{31\!\cdots\!78}a^{17}-\frac{66\!\cdots\!03}{31\!\cdots\!78}a^{16}+\frac{29\!\cdots\!56}{12\!\cdots\!53}a^{15}-\frac{54\!\cdots\!25}{31\!\cdots\!78}a^{14}-\frac{10\!\cdots\!86}{15\!\cdots\!89}a^{13}+\frac{13\!\cdots\!81}{31\!\cdots\!78}a^{12}-\frac{13\!\cdots\!60}{15\!\cdots\!89}a^{11}+\frac{18\!\cdots\!36}{15\!\cdots\!89}a^{10}-\frac{40\!\cdots\!37}{31\!\cdots\!78}a^{9}+\frac{33\!\cdots\!85}{31\!\cdots\!78}a^{8}-\frac{21\!\cdots\!85}{31\!\cdots\!78}a^{7}+\frac{50\!\cdots\!18}{15\!\cdots\!89}a^{6}-\frac{18\!\cdots\!82}{12\!\cdots\!53}a^{5}+\frac{14\!\cdots\!37}{15\!\cdots\!89}a^{4}-\frac{54\!\cdots\!28}{15\!\cdots\!89}a^{3}-\frac{24\!\cdots\!21}{15\!\cdots\!89}a^{2}+\frac{66\!\cdots\!92}{15\!\cdots\!89}a-\frac{45\!\cdots\!71}{15\!\cdots\!89}$, $\frac{13\!\cdots\!68}{15\!\cdots\!89}a^{23}+\frac{74\!\cdots\!39}{63\!\cdots\!56}a^{22}-\frac{51\!\cdots\!09}{63\!\cdots\!56}a^{21}+\frac{23\!\cdots\!39}{63\!\cdots\!56}a^{20}-\frac{82\!\cdots\!13}{63\!\cdots\!56}a^{19}+\frac{55\!\cdots\!59}{15\!\cdots\!89}a^{18}-\frac{24\!\cdots\!37}{31\!\cdots\!78}a^{17}+\frac{44\!\cdots\!11}{31\!\cdots\!78}a^{16}-\frac{48\!\cdots\!37}{24\!\cdots\!06}a^{15}+\frac{66\!\cdots\!21}{31\!\cdots\!78}a^{14}-\frac{17\!\cdots\!23}{15\!\cdots\!89}a^{13}-\frac{20\!\cdots\!47}{15\!\cdots\!89}a^{12}+\frac{74\!\cdots\!16}{15\!\cdots\!89}a^{11}-\frac{13\!\cdots\!39}{15\!\cdots\!89}a^{10}+\frac{34\!\cdots\!67}{31\!\cdots\!78}a^{9}-\frac{36\!\cdots\!03}{31\!\cdots\!78}a^{8}+\frac{30\!\cdots\!91}{31\!\cdots\!78}a^{7}-\frac{10\!\cdots\!87}{15\!\cdots\!89}a^{6}+\frac{42\!\cdots\!28}{12\!\cdots\!53}a^{5}-\frac{28\!\cdots\!68}{15\!\cdots\!89}a^{4}+\frac{14\!\cdots\!29}{15\!\cdots\!89}a^{3}-\frac{63\!\cdots\!86}{15\!\cdots\!89}a^{2}+\frac{15\!\cdots\!06}{15\!\cdots\!89}a-\frac{57\!\cdots\!06}{15\!\cdots\!89}$, $\frac{23\!\cdots\!02}{15\!\cdots\!89}a^{23}-\frac{54\!\cdots\!37}{31\!\cdots\!78}a^{22}+\frac{16\!\cdots\!46}{15\!\cdots\!89}a^{21}-\frac{12\!\cdots\!83}{31\!\cdots\!78}a^{20}+\frac{77\!\cdots\!95}{63\!\cdots\!56}a^{19}-\frac{92\!\cdots\!09}{31\!\cdots\!78}a^{18}+\frac{88\!\cdots\!89}{15\!\cdots\!89}a^{17}-\frac{26\!\cdots\!05}{31\!\cdots\!78}a^{16}+\frac{11\!\cdots\!79}{12\!\cdots\!53}a^{15}-\frac{92\!\cdots\!73}{15\!\cdots\!89}a^{14}-\frac{13\!\cdots\!11}{31\!\cdots\!78}a^{13}+\frac{60\!\cdots\!15}{31\!\cdots\!78}a^{12}-\frac{11\!\cdots\!75}{31\!\cdots\!78}a^{11}+\frac{14\!\cdots\!69}{31\!\cdots\!78}a^{10}-\frac{77\!\cdots\!74}{15\!\cdots\!89}a^{9}+\frac{61\!\cdots\!47}{15\!\cdots\!89}a^{8}-\frac{75\!\cdots\!05}{31\!\cdots\!78}a^{7}+\frac{15\!\cdots\!94}{15\!\cdots\!89}a^{6}-\frac{55\!\cdots\!27}{12\!\cdots\!53}a^{5}+\frac{42\!\cdots\!11}{15\!\cdots\!89}a^{4}-\frac{20\!\cdots\!97}{15\!\cdots\!89}a^{3}-\frac{18\!\cdots\!77}{15\!\cdots\!89}a^{2}+\frac{64\!\cdots\!15}{15\!\cdots\!89}a-\frac{62\!\cdots\!03}{15\!\cdots\!89}$, $\frac{10\!\cdots\!39}{31\!\cdots\!78}a^{23}+\frac{22\!\cdots\!59}{63\!\cdots\!56}a^{22}-\frac{29\!\cdots\!15}{15\!\cdots\!89}a^{21}+\frac{99\!\cdots\!09}{15\!\cdots\!89}a^{20}-\frac{98\!\cdots\!41}{63\!\cdots\!56}a^{19}+\frac{88\!\cdots\!65}{31\!\cdots\!78}a^{18}-\frac{10\!\cdots\!39}{31\!\cdots\!78}a^{17}+\frac{87\!\cdots\!91}{15\!\cdots\!89}a^{16}+\frac{19\!\cdots\!35}{24\!\cdots\!06}a^{15}-\frac{34\!\cdots\!11}{15\!\cdots\!89}a^{14}+\frac{10\!\cdots\!27}{31\!\cdots\!78}a^{13}-\frac{11\!\cdots\!45}{31\!\cdots\!78}a^{12}+\frac{63\!\cdots\!55}{31\!\cdots\!78}a^{11}+\frac{18\!\cdots\!38}{15\!\cdots\!89}a^{10}-\frac{81\!\cdots\!89}{15\!\cdots\!89}a^{9}+\frac{14\!\cdots\!87}{15\!\cdots\!89}a^{8}-\frac{31\!\cdots\!45}{31\!\cdots\!78}a^{7}+\frac{12\!\cdots\!02}{15\!\cdots\!89}a^{6}-\frac{50\!\cdots\!55}{12\!\cdots\!53}a^{5}+\frac{32\!\cdots\!43}{15\!\cdots\!89}a^{4}-\frac{24\!\cdots\!65}{15\!\cdots\!89}a^{3}+\frac{10\!\cdots\!73}{15\!\cdots\!89}a^{2}-\frac{34\!\cdots\!49}{15\!\cdots\!89}a+\frac{43\!\cdots\!25}{15\!\cdots\!89}$, $\frac{13\!\cdots\!27}{63\!\cdots\!56}a^{23}+\frac{14\!\cdots\!73}{63\!\cdots\!56}a^{22}-\frac{20\!\cdots\!33}{15\!\cdots\!89}a^{21}+\frac{78\!\cdots\!87}{15\!\cdots\!89}a^{20}-\frac{21\!\cdots\!45}{15\!\cdots\!89}a^{19}+\frac{96\!\cdots\!19}{31\!\cdots\!78}a^{18}-\frac{84\!\cdots\!30}{15\!\cdots\!89}a^{17}+\frac{22\!\cdots\!31}{31\!\cdots\!78}a^{16}-\frac{13\!\cdots\!89}{24\!\cdots\!06}a^{15}-\frac{13\!\cdots\!61}{31\!\cdots\!78}a^{14}+\frac{35\!\cdots\!05}{31\!\cdots\!78}a^{13}-\frac{36\!\cdots\!45}{15\!\cdots\!89}a^{12}+\frac{10\!\cdots\!19}{31\!\cdots\!78}a^{11}-\frac{10\!\cdots\!43}{31\!\cdots\!78}a^{10}+\frac{44\!\cdots\!93}{15\!\cdots\!89}a^{9}-\frac{19\!\cdots\!38}{15\!\cdots\!89}a^{8}-\frac{18\!\cdots\!76}{15\!\cdots\!89}a^{7}+\frac{68\!\cdots\!58}{15\!\cdots\!89}a^{6}-\frac{20\!\cdots\!32}{12\!\cdots\!53}a^{5}+\frac{11\!\cdots\!89}{15\!\cdots\!89}a^{4}-\frac{29\!\cdots\!50}{15\!\cdots\!89}a^{3}+\frac{11\!\cdots\!40}{15\!\cdots\!89}a^{2}-\frac{51\!\cdots\!00}{15\!\cdots\!89}a+\frac{44\!\cdots\!01}{15\!\cdots\!89}$
| 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: | \( 20547515.71936001 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 20547515.71936001 \cdot 1}{2\cdot\sqrt{1280000000000000000000000000000000}}\cr\approx \mathstrut & 0.440598492731830 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 12*x^23 + 74*x^22 - 308*x^21 + 966*x^20 - 2412*x^19 + 4884*x^18 - 7968*x^17 + 10016*x^16 - 8392*x^15 + 820*x^14 + 12680*x^13 - 29520*x^12 + 44840*x^11 - 53520*x^10 + 50544*x^9 - 38528*x^8 + 24016*x^7 - 13652*x^6 + 7544*x^5 - 3696*x^4 + 1472*x^3 - 384*x^2 + 88*x + 4)
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 - 12*x^23 + 74*x^22 - 308*x^21 + 966*x^20 - 2412*x^19 + 4884*x^18 - 7968*x^17 + 10016*x^16 - 8392*x^15 + 820*x^14 + 12680*x^13 - 29520*x^12 + 44840*x^11 - 53520*x^10 + 50544*x^9 - 38528*x^8 + 24016*x^7 - 13652*x^6 + 7544*x^5 - 3696*x^4 + 1472*x^3 - 384*x^2 + 88*x + 4, 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 - 12*x^23 + 74*x^22 - 308*x^21 + 966*x^20 - 2412*x^19 + 4884*x^18 - 7968*x^17 + 10016*x^16 - 8392*x^15 + 820*x^14 + 12680*x^13 - 29520*x^12 + 44840*x^11 - 53520*x^10 + 50544*x^9 - 38528*x^8 + 24016*x^7 - 13652*x^6 + 7544*x^5 - 3696*x^4 + 1472*x^3 - 384*x^2 + 88*x + 4);
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 - 12*x^23 + 74*x^22 - 308*x^21 + 966*x^20 - 2412*x^19 + 4884*x^18 - 7968*x^17 + 10016*x^16 - 8392*x^15 + 820*x^14 + 12680*x^13 - 29520*x^12 + 44840*x^11 - 53520*x^10 + 50544*x^9 - 38528*x^8 + 24016*x^7 - 13652*x^6 + 7544*x^5 - 3696*x^4 + 1472*x^3 - 384*x^2 + 88*x + 4);
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
$\GL(2,5)$ (as 24T1353):
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 480 |
| The 24 conjugacy class representatives for $\GL(2,5)$ |
| Character table for $\GL(2,5)$ |
Intermediate fields
| 6.2.5000000.1, 12.4.500000000000000.2 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $24$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{3}$ | $20{,}\,{\href{/padicField/19.4.0.1}{4} }$ | $24$ | ${\href{/padicField/29.12.0.1}{12} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{6}$ | $24$ | ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $24$ | $1$ | $38$ | |||
|
\(5\)
| 5.4.3.4 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| Deg $20$ | $5$ | $4$ | $28$ |