Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 66*x^22 - 8*x^21 + 1860*x^20 + 408*x^19 - 27668*x^18 - 7488*x^17 + 237498*x^16 + 58984*x^15 - 1240668*x^14 - 180132*x^13 + 4010886*x^12 + 120480*x^11 - 6473694*x^10 + 596120*x^9 + 3024492*x^8 - 1421952*x^7 - 2459420*x^6 - 3154656*x^5 + 26476854*x^4 + 21395960*x^3 - 17515860*x^2 - 7883700*x + 15732025)
gp: K = bnfinit(y^24 - 66*y^22 - 8*y^21 + 1860*y^20 + 408*y^19 - 27668*y^18 - 7488*y^17 + 237498*y^16 + 58984*y^15 - 1240668*y^14 - 180132*y^13 + 4010886*y^12 + 120480*y^11 - 6473694*y^10 + 596120*y^9 + 3024492*y^8 - 1421952*y^7 - 2459420*y^6 - 3154656*y^5 + 26476854*y^4 + 21395960*y^3 - 17515860*y^2 - 7883700*y + 15732025, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 66*x^22 - 8*x^21 + 1860*x^20 + 408*x^19 - 27668*x^18 - 7488*x^17 + 237498*x^16 + 58984*x^15 - 1240668*x^14 - 180132*x^13 + 4010886*x^12 + 120480*x^11 - 6473694*x^10 + 596120*x^9 + 3024492*x^8 - 1421952*x^7 - 2459420*x^6 - 3154656*x^5 + 26476854*x^4 + 21395960*x^3 - 17515860*x^2 - 7883700*x + 15732025);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 66*x^22 - 8*x^21 + 1860*x^20 + 408*x^19 - 27668*x^18 - 7488*x^17 + 237498*x^16 + 58984*x^15 - 1240668*x^14 - 180132*x^13 + 4010886*x^12 + 120480*x^11 - 6473694*x^10 + 596120*x^9 + 3024492*x^8 - 1421952*x^7 - 2459420*x^6 - 3154656*x^5 + 26476854*x^4 + 21395960*x^3 - 17515860*x^2 - 7883700*x + 15732025)
\( x^{24} - 66 x^{22} - 8 x^{21} + 1860 x^{20} + 408 x^{19} - 27668 x^{18} - 7488 x^{17} + 237498 x^{16} + \cdots + 15732025 \)
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: |
\(327347161010008025375825530454016000000000000\)
\(\medspace = 2^{36}\cdot 3^{28}\cdot 5^{12}\cdot 31^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(71.58\) | 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: | $2^{3/2}3^{7/6}5^{1/2}31^{1/2}\approx 126.86806219309365$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{12}a^{12}+\frac{1}{6}a^{9}+\frac{1}{6}a^{3}-\frac{1}{12}$, $\frac{1}{12}a^{13}+\frac{1}{6}a^{10}+\frac{1}{6}a^{4}-\frac{1}{12}a$, $\frac{1}{12}a^{14}+\frac{1}{6}a^{11}+\frac{1}{6}a^{5}-\frac{1}{12}a^{2}$, $\frac{1}{12}a^{15}+\frac{1}{6}a^{9}+\frac{1}{6}a^{6}+\frac{1}{12}a^{3}+\frac{1}{6}$, $\frac{1}{12}a^{16}+\frac{1}{6}a^{10}+\frac{1}{6}a^{7}+\frac{1}{12}a^{4}+\frac{1}{6}a$, $\frac{1}{12}a^{17}+\frac{1}{6}a^{11}+\frac{1}{6}a^{8}+\frac{1}{12}a^{5}+\frac{1}{6}a^{2}$, $\frac{1}{192}a^{18}-\frac{1}{48}a^{16}-\frac{1}{32}a^{15}-\frac{1}{48}a^{14}+\frac{1}{48}a^{13}+\frac{7}{192}a^{12}+\frac{1}{12}a^{11}-\frac{1}{8}a^{10}-\frac{3}{16}a^{9}-\frac{1}{8}a^{8}+\frac{5}{24}a^{7}+\frac{7}{192}a^{6}+\frac{1}{12}a^{5}+\frac{7}{48}a^{4}+\frac{7}{32}a^{3}-\frac{17}{48}a^{2}+\frac{7}{16}a+\frac{25}{192}$, $\frac{1}{192}a^{19}-\frac{1}{48}a^{17}-\frac{1}{32}a^{16}-\frac{1}{48}a^{15}+\frac{1}{48}a^{14}+\frac{7}{192}a^{13}-\frac{1}{8}a^{11}-\frac{3}{16}a^{10}+\frac{5}{24}a^{9}+\frac{5}{24}a^{8}+\frac{7}{192}a^{7}+\frac{1}{12}a^{6}+\frac{7}{48}a^{5}+\frac{7}{32}a^{4}-\frac{1}{48}a^{3}+\frac{7}{16}a^{2}+\frac{25}{192}a+\frac{1}{12}$, $\frac{1}{2880}a^{20}+\frac{1}{2880}a^{19}+\frac{1}{2880}a^{18}+\frac{3}{160}a^{17}+\frac{5}{288}a^{16}+\frac{41}{1440}a^{15}+\frac{13}{960}a^{14}-\frac{37}{2880}a^{13}-\frac{17}{576}a^{12}+\frac{133}{720}a^{11}+\frac{19}{80}a^{10}+\frac{31}{720}a^{9}-\frac{521}{2880}a^{8}-\frac{577}{2880}a^{7}+\frac{101}{960}a^{6}+\frac{683}{1440}a^{5}-\frac{607}{1440}a^{4}+\frac{13}{32}a^{3}+\frac{1001}{2880}a^{2}-\frac{247}{576}a+\frac{169}{576}$, $\frac{1}{2880}a^{21}-\frac{7}{2880}a^{18}-\frac{1}{720}a^{17}+\frac{1}{90}a^{16}+\frac{77}{2880}a^{15}-\frac{19}{720}a^{14}-\frac{1}{60}a^{13}-\frac{43}{2880}a^{12}+\frac{19}{360}a^{11}-\frac{7}{36}a^{10}+\frac{37}{192}a^{9}-\frac{7}{360}a^{8}-\frac{7}{36}a^{7}+\frac{163}{2880}a^{6}+\frac{5}{48}a^{5}-\frac{31}{180}a^{4}-\frac{529}{2880}a^{3}+\frac{161}{720}a^{2}+\frac{2}{9}a+\frac{59}{576}$, $\frac{1}{31\!\cdots\!80}a^{22}-\frac{36\!\cdots\!79}{62\!\cdots\!56}a^{21}+\frac{56\!\cdots\!53}{51\!\cdots\!80}a^{20}+\frac{74\!\cdots\!31}{31\!\cdots\!80}a^{19}+\frac{85\!\cdots\!87}{15\!\cdots\!40}a^{18}+\frac{73\!\cdots\!79}{19\!\cdots\!80}a^{17}+\frac{61\!\cdots\!97}{31\!\cdots\!80}a^{16}-\frac{23\!\cdots\!65}{62\!\cdots\!56}a^{15}+\frac{61\!\cdots\!57}{15\!\cdots\!40}a^{14}+\frac{54\!\cdots\!11}{31\!\cdots\!80}a^{13}+\frac{15\!\cdots\!63}{77\!\cdots\!20}a^{12}+\frac{12\!\cdots\!67}{64\!\cdots\!60}a^{11}+\frac{55\!\cdots\!67}{31\!\cdots\!80}a^{10}-\frac{59\!\cdots\!29}{31\!\cdots\!80}a^{9}-\frac{13\!\cdots\!99}{15\!\cdots\!40}a^{8}-\frac{58\!\cdots\!83}{31\!\cdots\!80}a^{7}+\frac{28\!\cdots\!07}{15\!\cdots\!40}a^{6}-\frac{34\!\cdots\!79}{97\!\cdots\!90}a^{5}-\frac{12\!\cdots\!41}{31\!\cdots\!80}a^{4}+\frac{75\!\cdots\!89}{31\!\cdots\!80}a^{3}+\frac{16\!\cdots\!93}{51\!\cdots\!80}a^{2}+\frac{19\!\cdots\!77}{62\!\cdots\!56}a-\frac{33\!\cdots\!95}{77\!\cdots\!32}$, $\frac{1}{70\!\cdots\!20}a^{23}+\frac{2388029122271}{70\!\cdots\!20}a^{22}-\frac{52\!\cdots\!17}{23\!\cdots\!40}a^{21}+\frac{58\!\cdots\!43}{70\!\cdots\!20}a^{20}+\frac{24\!\cdots\!43}{23\!\cdots\!40}a^{19}+\frac{18\!\cdots\!15}{23\!\cdots\!64}a^{18}+\frac{73\!\cdots\!11}{23\!\cdots\!40}a^{17}+\frac{31\!\cdots\!53}{83\!\cdots\!40}a^{16}+\frac{17\!\cdots\!19}{70\!\cdots\!20}a^{15}-\frac{78\!\cdots\!49}{14\!\cdots\!84}a^{14}+\frac{13\!\cdots\!13}{78\!\cdots\!80}a^{13}-\frac{46\!\cdots\!21}{17\!\cdots\!48}a^{12}+\frac{12\!\cdots\!03}{70\!\cdots\!20}a^{11}+\frac{82\!\cdots\!29}{70\!\cdots\!20}a^{10}+\frac{62\!\cdots\!91}{78\!\cdots\!80}a^{9}+\frac{37\!\cdots\!63}{23\!\cdots\!40}a^{8}+\frac{15\!\cdots\!43}{70\!\cdots\!20}a^{7}-\frac{31\!\cdots\!07}{35\!\cdots\!60}a^{6}+\frac{31\!\cdots\!67}{15\!\cdots\!76}a^{5}+\frac{45\!\cdots\!47}{23\!\cdots\!40}a^{4}+\frac{26\!\cdots\!73}{70\!\cdots\!20}a^{3}+\frac{20\!\cdots\!89}{78\!\cdots\!80}a^{2}-\frac{18\!\cdots\!45}{14\!\cdots\!84}a-\frac{18\!\cdots\!11}{11\!\cdots\!32}$
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
$C_{6}\times C_{12}\times C_{24}$, which has order $1728$ (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: |
\( -\frac{156298929093}{52910380044793504165} a^{23} - \frac{15721523105549}{338626432286678426656} a^{22} + \frac{1026775033870493}{5079396484300176399840} a^{21} + \frac{3930978028222657}{1269849121075044099960} a^{20} - \frac{14248126036067743}{2539698242150088199920} a^{19} - \frac{59595815324878733}{677252864573356853312} a^{18} + \frac{1627044830288313}{21164152017917401666} a^{17} + \frac{6682572261272587229}{5079396484300176399840} a^{16} - \frac{361894086914734207}{634924560537522049980} a^{15} - \frac{9520436810252697731}{846566080716696066640} a^{14} + \frac{2412148015466457751}{846566080716696066640} a^{13} + \frac{582610218259638597007}{10158792968600352799680} a^{12} - \frac{1763547714117624329}{126984912107504409996} a^{11} - \frac{884910100261156878409}{5079396484300176399840} a^{10} + \frac{234890694011930026873}{5079396484300176399840} a^{9} + \frac{143909127786311489387}{634924560537522049980} a^{8} - \frac{167597872326918016577}{2539698242150088199920} a^{7} + \frac{917834418066691851491}{10158792968600352799680} a^{6} - \frac{15973400743667903527}{634924560537522049980} a^{5} - \frac{173075723995518428131}{1693132161433392133280} a^{4} + \frac{477638837536008091189}{2539698242150088199920} a^{3} - \frac{3085670818197919309133}{2539698242150088199920} a^{2} - \frac{105912593222289653195}{169313216143339213328} a + \frac{2155872873194313877541}{2031758593720070559936} \)
(order $6$)
| 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: |
$\frac{22\!\cdots\!02}{54\!\cdots\!05}a^{23}-\frac{16\!\cdots\!43}{25\!\cdots\!40}a^{22}-\frac{13\!\cdots\!31}{51\!\cdots\!88}a^{21}+\frac{54\!\cdots\!25}{12\!\cdots\!72}a^{20}+\frac{99\!\cdots\!57}{12\!\cdots\!20}a^{19}-\frac{60\!\cdots\!03}{51\!\cdots\!80}a^{18}-\frac{68\!\cdots\!32}{54\!\cdots\!05}a^{17}+\frac{45\!\cdots\!33}{25\!\cdots\!40}a^{16}+\frac{13\!\cdots\!11}{10\!\cdots\!10}a^{15}-\frac{19\!\cdots\!93}{12\!\cdots\!20}a^{14}-\frac{92\!\cdots\!39}{12\!\cdots\!20}a^{13}+\frac{39\!\cdots\!03}{51\!\cdots\!80}a^{12}+\frac{19\!\cdots\!41}{10\!\cdots\!10}a^{11}-\frac{20\!\cdots\!11}{86\!\cdots\!80}a^{10}-\frac{69\!\cdots\!99}{25\!\cdots\!40}a^{9}+\frac{49\!\cdots\!66}{16\!\cdots\!15}a^{8}-\frac{74\!\cdots\!09}{12\!\cdots\!20}a^{7}+\frac{23\!\cdots\!81}{34\!\cdots\!92}a^{6}-\frac{11\!\cdots\!41}{10\!\cdots\!10}a^{5}+\frac{44\!\cdots\!83}{25\!\cdots\!40}a^{4}+\frac{16\!\cdots\!23}{43\!\cdots\!40}a^{3}-\frac{28\!\cdots\!09}{12\!\cdots\!20}a^{2}-\frac{25\!\cdots\!09}{25\!\cdots\!44}a+\frac{71\!\cdots\!55}{34\!\cdots\!92}$, $\frac{61\!\cdots\!37}{74\!\cdots\!80}a^{23}+\frac{13\!\cdots\!43}{14\!\cdots\!60}a^{22}-\frac{12\!\cdots\!21}{22\!\cdots\!40}a^{21}-\frac{36\!\cdots\!03}{56\!\cdots\!60}a^{20}+\frac{66\!\cdots\!79}{44\!\cdots\!80}a^{19}+\frac{29\!\cdots\!29}{14\!\cdots\!60}a^{18}-\frac{32\!\cdots\!49}{14\!\cdots\!96}a^{17}-\frac{14\!\cdots\!83}{44\!\cdots\!80}a^{16}+\frac{19\!\cdots\!99}{11\!\cdots\!20}a^{15}+\frac{49\!\cdots\!81}{18\!\cdots\!20}a^{14}-\frac{12\!\cdots\!39}{14\!\cdots\!60}a^{13}-\frac{57\!\cdots\!71}{44\!\cdots\!80}a^{12}+\frac{56\!\cdots\!93}{22\!\cdots\!40}a^{11}+\frac{16\!\cdots\!63}{44\!\cdots\!80}a^{10}-\frac{70\!\cdots\!37}{22\!\cdots\!40}a^{9}-\frac{28\!\cdots\!59}{56\!\cdots\!60}a^{8}-\frac{23\!\cdots\!63}{44\!\cdots\!80}a^{7}-\frac{42\!\cdots\!43}{44\!\cdots\!80}a^{6}+\frac{10\!\cdots\!79}{22\!\cdots\!40}a^{5}-\frac{18\!\cdots\!87}{14\!\cdots\!60}a^{4}+\frac{98\!\cdots\!73}{56\!\cdots\!60}a^{3}+\frac{20\!\cdots\!79}{56\!\cdots\!60}a^{2}+\frac{36\!\cdots\!15}{29\!\cdots\!92}a-\frac{98\!\cdots\!93}{89\!\cdots\!76}$, $\frac{47\!\cdots\!27}{24\!\cdots\!90}a^{23}-\frac{14\!\cdots\!43}{78\!\cdots\!80}a^{22}-\frac{12\!\cdots\!56}{12\!\cdots\!45}a^{21}+\frac{12\!\cdots\!39}{11\!\cdots\!32}a^{20}+\frac{11\!\cdots\!91}{58\!\cdots\!60}a^{19}-\frac{15\!\cdots\!23}{58\!\cdots\!60}a^{18}-\frac{76\!\cdots\!07}{11\!\cdots\!32}a^{17}+\frac{31\!\cdots\!87}{78\!\cdots\!80}a^{16}-\frac{87\!\cdots\!41}{29\!\cdots\!80}a^{15}-\frac{11\!\cdots\!17}{29\!\cdots\!80}a^{14}+\frac{18\!\cdots\!09}{39\!\cdots\!40}a^{13}+\frac{19\!\cdots\!97}{97\!\cdots\!60}a^{12}-\frac{94\!\cdots\!87}{29\!\cdots\!80}a^{11}-\frac{38\!\cdots\!51}{23\!\cdots\!40}a^{10}+\frac{17\!\cdots\!71}{14\!\cdots\!54}a^{9}-\frac{55\!\cdots\!83}{19\!\cdots\!20}a^{8}-\frac{59\!\cdots\!53}{29\!\cdots\!08}a^{7}+\frac{49\!\cdots\!01}{58\!\cdots\!60}a^{6}-\frac{46\!\cdots\!89}{39\!\cdots\!44}a^{5}+\frac{20\!\cdots\!27}{46\!\cdots\!28}a^{4}+\frac{32\!\cdots\!43}{97\!\cdots\!60}a^{3}-\frac{42\!\cdots\!49}{14\!\cdots\!40}a^{2}-\frac{26\!\cdots\!05}{83\!\cdots\!44}a+\frac{16\!\cdots\!49}{48\!\cdots\!18}$, $\frac{30\!\cdots\!71}{23\!\cdots\!64}a^{23}-\frac{22\!\cdots\!99}{23\!\cdots\!40}a^{22}-\frac{98\!\cdots\!83}{11\!\cdots\!20}a^{21}+\frac{31\!\cdots\!37}{58\!\cdots\!16}a^{20}+\frac{55\!\cdots\!03}{23\!\cdots\!40}a^{19}-\frac{30\!\cdots\!07}{23\!\cdots\!40}a^{18}-\frac{40\!\cdots\!27}{11\!\cdots\!20}a^{17}+\frac{27\!\cdots\!75}{15\!\cdots\!76}a^{16}+\frac{16\!\cdots\!83}{58\!\cdots\!60}a^{15}-\frac{58\!\cdots\!98}{36\!\cdots\!35}a^{14}-\frac{68\!\cdots\!33}{46\!\cdots\!28}a^{13}+\frac{47\!\cdots\!79}{46\!\cdots\!28}a^{12}+\frac{16\!\cdots\!63}{39\!\cdots\!40}a^{11}-\frac{58\!\cdots\!03}{15\!\cdots\!76}a^{10}-\frac{53\!\cdots\!23}{11\!\cdots\!20}a^{9}+\frac{19\!\cdots\!27}{29\!\cdots\!80}a^{8}-\frac{28\!\cdots\!29}{78\!\cdots\!80}a^{7}-\frac{25\!\cdots\!11}{78\!\cdots\!80}a^{6}+\frac{18\!\cdots\!47}{11\!\cdots\!20}a^{5}-\frac{18\!\cdots\!33}{23\!\cdots\!40}a^{4}+\frac{64\!\cdots\!63}{14\!\cdots\!40}a^{3}-\frac{20\!\cdots\!06}{36\!\cdots\!35}a^{2}-\frac{12\!\cdots\!39}{46\!\cdots\!28}a+\frac{10\!\cdots\!57}{46\!\cdots\!28}$, $\frac{10\!\cdots\!91}{11\!\cdots\!20}a^{23}-\frac{51\!\cdots\!93}{39\!\cdots\!40}a^{22}-\frac{58\!\cdots\!63}{97\!\cdots\!60}a^{21}+\frac{21\!\cdots\!03}{58\!\cdots\!60}a^{20}+\frac{12\!\cdots\!93}{78\!\cdots\!80}a^{19}+\frac{26\!\cdots\!61}{14\!\cdots\!40}a^{18}-\frac{18\!\cdots\!67}{78\!\cdots\!88}a^{17}-\frac{25\!\cdots\!43}{58\!\cdots\!60}a^{16}+\frac{71\!\cdots\!27}{39\!\cdots\!44}a^{15}+\frac{73\!\cdots\!17}{19\!\cdots\!20}a^{14}-\frac{19\!\cdots\!83}{23\!\cdots\!40}a^{13}-\frac{12\!\cdots\!27}{97\!\cdots\!36}a^{12}+\frac{74\!\cdots\!23}{39\!\cdots\!40}a^{11}+\frac{14\!\cdots\!67}{39\!\cdots\!40}a^{10}+\frac{10\!\cdots\!77}{14\!\cdots\!40}a^{9}-\frac{99\!\cdots\!07}{58\!\cdots\!60}a^{8}-\frac{17\!\cdots\!71}{23\!\cdots\!40}a^{7}+\frac{58\!\cdots\!43}{14\!\cdots\!54}a^{6}+\frac{46\!\cdots\!93}{11\!\cdots\!20}a^{5}-\frac{16\!\cdots\!29}{29\!\cdots\!80}a^{4}+\frac{12\!\cdots\!69}{58\!\cdots\!60}a^{3}+\frac{23\!\cdots\!39}{19\!\cdots\!20}a^{2}+\frac{21\!\cdots\!25}{15\!\cdots\!76}a+\frac{28\!\cdots\!73}{73\!\cdots\!27}$, $\frac{13\!\cdots\!21}{23\!\cdots\!40}a^{23}+\frac{85\!\cdots\!53}{70\!\cdots\!20}a^{22}-\frac{24\!\cdots\!21}{70\!\cdots\!92}a^{21}-\frac{33\!\cdots\!73}{23\!\cdots\!40}a^{20}+\frac{62\!\cdots\!79}{70\!\cdots\!20}a^{19}+\frac{43\!\cdots\!69}{78\!\cdots\!80}a^{18}-\frac{51\!\cdots\!31}{46\!\cdots\!28}a^{17}-\frac{22\!\cdots\!29}{23\!\cdots\!40}a^{16}+\frac{19\!\cdots\!23}{29\!\cdots\!80}a^{15}+\frac{11\!\cdots\!65}{14\!\cdots\!84}a^{14}-\frac{10\!\cdots\!17}{70\!\cdots\!20}a^{13}-\frac{94\!\cdots\!03}{23\!\cdots\!40}a^{12}-\frac{23\!\cdots\!11}{70\!\cdots\!20}a^{11}+\frac{18\!\cdots\!47}{14\!\cdots\!84}a^{10}+\frac{23\!\cdots\!23}{70\!\cdots\!92}a^{9}-\frac{54\!\cdots\!51}{23\!\cdots\!40}a^{8}-\frac{23\!\cdots\!69}{46\!\cdots\!28}a^{7}-\frac{11\!\cdots\!17}{70\!\cdots\!20}a^{6}-\frac{25\!\cdots\!63}{70\!\cdots\!20}a^{5}+\frac{17\!\cdots\!29}{23\!\cdots\!40}a^{4}+\frac{97\!\cdots\!97}{19\!\cdots\!20}a^{3}+\frac{17\!\cdots\!07}{70\!\cdots\!20}a^{2}+\frac{20\!\cdots\!23}{46\!\cdots\!28}a+\frac{41\!\cdots\!17}{14\!\cdots\!84}$, $\frac{11\!\cdots\!39}{78\!\cdots\!80}a^{23}-\frac{15\!\cdots\!73}{35\!\cdots\!60}a^{22}-\frac{41\!\cdots\!09}{43\!\cdots\!62}a^{21}+\frac{53\!\cdots\!21}{29\!\cdots\!80}a^{20}+\frac{19\!\cdots\!09}{70\!\cdots\!20}a^{19}-\frac{32\!\cdots\!83}{11\!\cdots\!20}a^{18}-\frac{97\!\cdots\!37}{23\!\cdots\!40}a^{17}+\frac{52\!\cdots\!67}{19\!\cdots\!20}a^{16}+\frac{14\!\cdots\!93}{39\!\cdots\!44}a^{15}-\frac{12\!\cdots\!73}{35\!\cdots\!60}a^{14}-\frac{14\!\cdots\!97}{70\!\cdots\!20}a^{13}+\frac{47\!\cdots\!13}{11\!\cdots\!20}a^{12}+\frac{48\!\cdots\!57}{70\!\cdots\!20}a^{11}-\frac{74\!\cdots\!61}{35\!\cdots\!60}a^{10}-\frac{20\!\cdots\!25}{17\!\cdots\!48}a^{9}+\frac{24\!\cdots\!41}{58\!\cdots\!60}a^{8}+\frac{22\!\cdots\!29}{46\!\cdots\!28}a^{7}+\frac{75\!\cdots\!41}{35\!\cdots\!60}a^{6}+\frac{32\!\cdots\!23}{70\!\cdots\!20}a^{5}-\frac{19\!\cdots\!83}{97\!\cdots\!60}a^{4}+\frac{20\!\cdots\!23}{58\!\cdots\!60}a^{3}+\frac{51\!\cdots\!43}{35\!\cdots\!60}a^{2}-\frac{18\!\cdots\!21}{46\!\cdots\!28}a+\frac{17\!\cdots\!33}{70\!\cdots\!92}$, $\frac{80\!\cdots\!89}{23\!\cdots\!40}a^{23}+\frac{79\!\cdots\!27}{70\!\cdots\!20}a^{22}-\frac{15\!\cdots\!87}{70\!\cdots\!20}a^{21}-\frac{25\!\cdots\!03}{70\!\cdots\!20}a^{20}+\frac{11\!\cdots\!71}{17\!\cdots\!80}a^{19}+\frac{12\!\cdots\!51}{70\!\cdots\!20}a^{18}-\frac{44\!\cdots\!07}{46\!\cdots\!28}a^{17}-\frac{22\!\cdots\!87}{70\!\cdots\!20}a^{16}+\frac{11\!\cdots\!33}{14\!\cdots\!84}a^{15}+\frac{19\!\cdots\!69}{70\!\cdots\!20}a^{14}-\frac{48\!\cdots\!13}{11\!\cdots\!20}a^{13}-\frac{16\!\cdots\!09}{14\!\cdots\!84}a^{12}+\frac{92\!\cdots\!33}{70\!\cdots\!20}a^{11}+\frac{20\!\cdots\!77}{70\!\cdots\!20}a^{10}-\frac{48\!\cdots\!31}{23\!\cdots\!40}a^{9}-\frac{32\!\cdots\!33}{70\!\cdots\!20}a^{8}+\frac{65\!\cdots\!03}{87\!\cdots\!40}a^{7}+\frac{25\!\cdots\!61}{14\!\cdots\!84}a^{6}-\frac{38\!\cdots\!59}{70\!\cdots\!20}a^{5}-\frac{41\!\cdots\!77}{70\!\cdots\!20}a^{4}+\frac{18\!\cdots\!97}{23\!\cdots\!40}a^{3}+\frac{68\!\cdots\!07}{78\!\cdots\!80}a^{2}-\frac{31\!\cdots\!75}{70\!\cdots\!92}a-\frac{14\!\cdots\!25}{19\!\cdots\!44}$, $\frac{34\!\cdots\!07}{78\!\cdots\!80}a^{23}-\frac{92\!\cdots\!73}{51\!\cdots\!80}a^{22}-\frac{28\!\cdots\!93}{97\!\cdots\!60}a^{21}+\frac{54\!\cdots\!27}{70\!\cdots\!20}a^{20}+\frac{58\!\cdots\!73}{70\!\cdots\!92}a^{19}-\frac{41\!\cdots\!73}{35\!\cdots\!60}a^{18}-\frac{96\!\cdots\!73}{78\!\cdots\!80}a^{17}+\frac{47\!\cdots\!31}{70\!\cdots\!20}a^{16}+\frac{18\!\cdots\!43}{17\!\cdots\!80}a^{15}-\frac{34\!\cdots\!93}{15\!\cdots\!76}a^{14}-\frac{48\!\cdots\!77}{87\!\cdots\!40}a^{13}+\frac{90\!\cdots\!67}{35\!\cdots\!60}a^{12}+\frac{12\!\cdots\!09}{70\!\cdots\!20}a^{11}-\frac{24\!\cdots\!67}{23\!\cdots\!40}a^{10}-\frac{58\!\cdots\!53}{21\!\cdots\!10}a^{9}-\frac{14\!\cdots\!43}{70\!\cdots\!20}a^{8}+\frac{32\!\cdots\!29}{35\!\cdots\!60}a^{7}+\frac{89\!\cdots\!03}{11\!\cdots\!20}a^{6}-\frac{22\!\cdots\!67}{70\!\cdots\!20}a^{5}+\frac{59\!\cdots\!89}{70\!\cdots\!20}a^{4}+\frac{67\!\cdots\!33}{58\!\cdots\!60}a^{3}+\frac{71\!\cdots\!09}{70\!\cdots\!20}a^{2}-\frac{22\!\cdots\!77}{35\!\cdots\!96}a-\frac{55\!\cdots\!19}{70\!\cdots\!92}$, $\frac{12\!\cdots\!09}{15\!\cdots\!76}a^{23}+\frac{18\!\cdots\!85}{14\!\cdots\!84}a^{22}-\frac{30\!\cdots\!71}{87\!\cdots\!40}a^{21}-\frac{13\!\cdots\!31}{15\!\cdots\!76}a^{20}+\frac{14\!\cdots\!39}{70\!\cdots\!92}a^{19}+\frac{92\!\cdots\!07}{39\!\cdots\!40}a^{18}+\frac{12\!\cdots\!73}{78\!\cdots\!80}a^{17}-\frac{26\!\cdots\!69}{78\!\cdots\!80}a^{16}-\frac{73\!\cdots\!41}{19\!\cdots\!20}a^{15}+\frac{18\!\cdots\!53}{70\!\cdots\!20}a^{14}+\frac{63\!\cdots\!91}{17\!\cdots\!80}a^{13}-\frac{47\!\cdots\!17}{39\!\cdots\!40}a^{12}-\frac{12\!\cdots\!61}{70\!\cdots\!20}a^{11}+\frac{45\!\cdots\!51}{14\!\cdots\!84}a^{10}+\frac{47\!\cdots\!55}{87\!\cdots\!24}a^{9}-\frac{13\!\cdots\!03}{51\!\cdots\!80}a^{8}-\frac{16\!\cdots\!03}{23\!\cdots\!64}a^{7}-\frac{13\!\cdots\!67}{35\!\cdots\!60}a^{6}-\frac{26\!\cdots\!49}{14\!\cdots\!84}a^{5}-\frac{12\!\cdots\!89}{23\!\cdots\!40}a^{4}-\frac{15\!\cdots\!39}{58\!\cdots\!60}a^{3}+\frac{27\!\cdots\!63}{70\!\cdots\!20}a^{2}+\frac{12\!\cdots\!09}{19\!\cdots\!72}a+\frac{24\!\cdots\!37}{70\!\cdots\!92}$, $\frac{38\!\cdots\!01}{11\!\cdots\!20}a^{23}+\frac{45\!\cdots\!91}{23\!\cdots\!40}a^{22}-\frac{85\!\cdots\!67}{39\!\cdots\!40}a^{21}-\frac{11\!\cdots\!25}{70\!\cdots\!92}a^{20}+\frac{43\!\cdots\!57}{70\!\cdots\!20}a^{19}+\frac{36\!\cdots\!49}{70\!\cdots\!20}a^{18}-\frac{70\!\cdots\!37}{78\!\cdots\!88}a^{17}-\frac{60\!\cdots\!47}{70\!\cdots\!20}a^{16}+\frac{13\!\cdots\!39}{17\!\cdots\!80}a^{15}+\frac{90\!\cdots\!07}{11\!\cdots\!20}a^{14}-\frac{28\!\cdots\!03}{70\!\cdots\!20}a^{13}-\frac{27\!\cdots\!81}{70\!\cdots\!20}a^{12}+\frac{47\!\cdots\!81}{35\!\cdots\!60}a^{11}+\frac{11\!\cdots\!17}{98\!\cdots\!20}a^{10}-\frac{16\!\cdots\!71}{70\!\cdots\!92}a^{9}-\frac{68\!\cdots\!19}{35\!\cdots\!60}a^{8}+\frac{20\!\cdots\!23}{14\!\cdots\!84}a^{7}+\frac{19\!\cdots\!49}{23\!\cdots\!40}a^{6}-\frac{68\!\cdots\!87}{70\!\cdots\!92}a^{5}-\frac{16\!\cdots\!37}{14\!\cdots\!84}a^{4}+\frac{29\!\cdots\!03}{36\!\cdots\!35}a^{3}+\frac{47\!\cdots\!39}{35\!\cdots\!60}a^{2}-\frac{37\!\cdots\!41}{14\!\cdots\!84}a-\frac{20\!\cdots\!91}{14\!\cdots\!84}$
| 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: | \( 19752228025.7156 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 19752228025.7156 \cdot 1728}{6\cdot\sqrt{327347161010008025375825530454016000000000000}}\cr\approx \mathstrut & 1.19031720024269 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 66*x^22 - 8*x^21 + 1860*x^20 + 408*x^19 - 27668*x^18 - 7488*x^17 + 237498*x^16 + 58984*x^15 - 1240668*x^14 - 180132*x^13 + 4010886*x^12 + 120480*x^11 - 6473694*x^10 + 596120*x^9 + 3024492*x^8 - 1421952*x^7 - 2459420*x^6 - 3154656*x^5 + 26476854*x^4 + 21395960*x^3 - 17515860*x^2 - 7883700*x + 15732025)
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 - 66*x^22 - 8*x^21 + 1860*x^20 + 408*x^19 - 27668*x^18 - 7488*x^17 + 237498*x^16 + 58984*x^15 - 1240668*x^14 - 180132*x^13 + 4010886*x^12 + 120480*x^11 - 6473694*x^10 + 596120*x^9 + 3024492*x^8 - 1421952*x^7 - 2459420*x^6 - 3154656*x^5 + 26476854*x^4 + 21395960*x^3 - 17515860*x^2 - 7883700*x + 15732025, 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 - 66*x^22 - 8*x^21 + 1860*x^20 + 408*x^19 - 27668*x^18 - 7488*x^17 + 237498*x^16 + 58984*x^15 - 1240668*x^14 - 180132*x^13 + 4010886*x^12 + 120480*x^11 - 6473694*x^10 + 596120*x^9 + 3024492*x^8 - 1421952*x^7 - 2459420*x^6 - 3154656*x^5 + 26476854*x^4 + 21395960*x^3 - 17515860*x^2 - 7883700*x + 15732025);
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 - 66*x^22 - 8*x^21 + 1860*x^20 + 408*x^19 - 27668*x^18 - 7488*x^17 + 237498*x^16 + 58984*x^15 - 1240668*x^14 - 180132*x^13 + 4010886*x^12 + 120480*x^11 - 6473694*x^10 + 596120*x^9 + 3024492*x^8 - 1421952*x^7 - 2459420*x^6 - 3154656*x^5 + 26476854*x^4 + 21395960*x^3 - 17515860*x^2 - 7883700*x + 15732025);
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
$C_2^2\times D_6$ (as 24T30):
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 48 |
| The 24 conjugacy class representatives for $C_2^2\times D_6$ |
| Character table for $C_2^2\times D_6$ is not computed |
Intermediate fields
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 24 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 | R | R | R | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{8}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.2.0.1}{2} }^{12}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(3\)
| 3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ |
| 3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |