Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 - 74*x^22 + 572*x^21 + 3016*x^20 - 17760*x^19 - 85518*x^18 + 285704*x^17 + 1661305*x^16 - 1906632*x^15 - 20467124*x^14 - 10151352*x^13 + 139734456*x^12 + 275550256*x^11 - 290937638*x^10 - 1685333160*x^9 - 1899102168*x^8 + 1612921144*x^7 + 7900602944*x^6 + 12266775908*x^5 + 11114510270*x^4 + 6111392440*x^3 + 2347621720*x^2 + 1082467988*x + 466788641)
gp: K = bnfinit(y^24 - 8*y^23 - 74*y^22 + 572*y^21 + 3016*y^20 - 17760*y^19 - 85518*y^18 + 285704*y^17 + 1661305*y^16 - 1906632*y^15 - 20467124*y^14 - 10151352*y^13 + 139734456*y^12 + 275550256*y^11 - 290937638*y^10 - 1685333160*y^9 - 1899102168*y^8 + 1612921144*y^7 + 7900602944*y^6 + 12266775908*y^5 + 11114510270*y^4 + 6111392440*y^3 + 2347621720*y^2 + 1082467988*y + 466788641, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 8*x^23 - 74*x^22 + 572*x^21 + 3016*x^20 - 17760*x^19 - 85518*x^18 + 285704*x^17 + 1661305*x^16 - 1906632*x^15 - 20467124*x^14 - 10151352*x^13 + 139734456*x^12 + 275550256*x^11 - 290937638*x^10 - 1685333160*x^9 - 1899102168*x^8 + 1612921144*x^7 + 7900602944*x^6 + 12266775908*x^5 + 11114510270*x^4 + 6111392440*x^3 + 2347621720*x^2 + 1082467988*x + 466788641);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 8*x^23 - 74*x^22 + 572*x^21 + 3016*x^20 - 17760*x^19 - 85518*x^18 + 285704*x^17 + 1661305*x^16 - 1906632*x^15 - 20467124*x^14 - 10151352*x^13 + 139734456*x^12 + 275550256*x^11 - 290937638*x^10 - 1685333160*x^9 - 1899102168*x^8 + 1612921144*x^7 + 7900602944*x^6 + 12266775908*x^5 + 11114510270*x^4 + 6111392440*x^3 + 2347621720*x^2 + 1082467988*x + 466788641)
\( x^{24} - 8 x^{23} - 74 x^{22} + 572 x^{21} + 3016 x^{20} - 17760 x^{19} - 85518 x^{18} + \cdots + 466788641 \)
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: |
\(3234204723240544858872018632704000000000000000000\)
\(\medspace = 2^{36}\cdot 5^{18}\cdot 37^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(105.01\) | 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: | $2^{3/2}5^{3/4}37^{2/3}\approx 105.01233438533802$ | ||
| Ramified primes: |
\(2\), \(5\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1480=2^{3}\cdot 5\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1480}(1,·)$, $\chi_{1480}(963,·)$, $\chi_{1480}(137,·)$, $\chi_{1480}(1099,·)$, $\chi_{1480}(593,·)$, $\chi_{1480}(787,·)$, $\chi_{1480}(121,·)$, $\chi_{1480}(729,·)$, $\chi_{1480}(667,·)$, $\chi_{1480}(1379,·)$, $\chi_{1480}(491,·)$, $\chi_{1480}(417,·)$, $\chi_{1480}(803,·)$, $\chi_{1480}(1009,·)$, $\chi_{1480}(297,·)$, $\chi_{1480}(1259,·)$, $\chi_{1480}(433,·)$, $\chi_{1480}(211,·)$, $\chi_{1480}(713,·)$, $\chi_{1480}(371,·)$, $\chi_{1480}(889,·)$, $\chi_{1480}(1083,·)$, $\chi_{1480}(1321,·)$, $\chi_{1480}(507,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{22}a^{11}-\frac{1}{2}a^{3}-\frac{1}{22}a$, $\frac{1}{44}a^{12}-\frac{1}{4}a^{8}+\frac{5}{22}a^{2}-\frac{1}{4}$, $\frac{1}{44}a^{13}-\frac{1}{4}a^{9}+\frac{5}{22}a^{3}-\frac{1}{4}a$, $\frac{1}{44}a^{14}-\frac{1}{4}a^{10}+\frac{5}{22}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{44}a^{15}-\frac{1}{44}a^{11}+\frac{5}{22}a^{5}+\frac{1}{4}a^{3}-\frac{5}{22}a$, $\frac{1}{44}a^{16}-\frac{1}{4}a^{8}+\frac{5}{22}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{44}a^{17}-\frac{1}{4}a^{9}+\frac{5}{22}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a$, $\frac{1}{88}a^{18}-\frac{1}{88}a^{16}-\frac{1}{88}a^{14}-\frac{1}{4}a^{10}-\frac{1}{88}a^{8}-\frac{21}{88}a^{6}-\frac{21}{88}a^{4}-\frac{1}{8}$, $\frac{1}{968}a^{19}+\frac{1}{484}a^{18}-\frac{3}{968}a^{17}+\frac{1}{484}a^{16}+\frac{3}{968}a^{15}-\frac{1}{121}a^{14}-\frac{1}{242}a^{13}+\frac{1}{242}a^{12}+\frac{7}{484}a^{11}-\frac{1}{44}a^{10}+\frac{109}{968}a^{9}-\frac{89}{484}a^{8}+\frac{91}{968}a^{7}-\frac{23}{484}a^{6}+\frac{261}{968}a^{5}-\frac{183}{484}a^{4}+\frac{39}{121}a^{3}+\frac{163}{484}a^{2}+\frac{63}{968}a-\frac{9}{44}$, $\frac{1}{968}a^{20}+\frac{1}{242}a^{18}+\frac{1}{121}a^{17}+\frac{5}{484}a^{16}+\frac{1}{121}a^{15}+\frac{1}{968}a^{14}-\frac{5}{484}a^{13}+\frac{3}{484}a^{12}+\frac{2}{121}a^{11}-\frac{89}{968}a^{10}-\frac{7}{44}a^{9}+\frac{97}{484}a^{8}-\frac{57}{242}a^{7}-\frac{71}{484}a^{6}+\frac{75}{242}a^{5}-\frac{397}{968}a^{4}-\frac{69}{242}a^{3}+\frac{379}{968}a^{2}-\frac{195}{484}a-\frac{41}{88}$, $\frac{1}{968}a^{21}-\frac{1}{88}a^{15}-\frac{13}{968}a^{11}+\frac{2}{11}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{23}{88}a^{5}-\frac{1}{8}a^{3}-\frac{351}{968}a-\frac{2}{11}$, $\frac{1}{16\!\cdots\!68}a^{22}-\frac{22\!\cdots\!59}{16\!\cdots\!68}a^{21}+\frac{39\!\cdots\!73}{83\!\cdots\!84}a^{20}-\frac{46\!\cdots\!77}{16\!\cdots\!68}a^{19}-\frac{10\!\cdots\!47}{20\!\cdots\!21}a^{18}+\frac{11\!\cdots\!57}{16\!\cdots\!68}a^{17}-\frac{11\!\cdots\!19}{16\!\cdots\!68}a^{16}-\frac{90\!\cdots\!61}{83\!\cdots\!84}a^{15}+\frac{14\!\cdots\!11}{83\!\cdots\!84}a^{14}+\frac{27\!\cdots\!19}{41\!\cdots\!42}a^{13}-\frac{10\!\cdots\!73}{16\!\cdots\!68}a^{12}-\frac{22\!\cdots\!59}{16\!\cdots\!68}a^{11}-\frac{36\!\cdots\!16}{20\!\cdots\!21}a^{10}+\frac{46\!\cdots\!11}{16\!\cdots\!68}a^{9}-\frac{93\!\cdots\!73}{83\!\cdots\!84}a^{8}-\frac{37\!\cdots\!41}{16\!\cdots\!68}a^{7}-\frac{19\!\cdots\!11}{16\!\cdots\!68}a^{6}+\frac{15\!\cdots\!77}{83\!\cdots\!84}a^{5}+\frac{30\!\cdots\!41}{16\!\cdots\!68}a^{4}-\frac{34\!\cdots\!69}{16\!\cdots\!68}a^{3}-\frac{49\!\cdots\!21}{16\!\cdots\!68}a^{2}-\frac{20\!\cdots\!91}{20\!\cdots\!21}a-\frac{36\!\cdots\!73}{76\!\cdots\!44}$, $\frac{1}{92\!\cdots\!88}a^{23}+\frac{4749054720471}{23\!\cdots\!22}a^{22}-\frac{32\!\cdots\!07}{11\!\cdots\!61}a^{21}-\frac{16\!\cdots\!69}{46\!\cdots\!44}a^{20}+\frac{11\!\cdots\!67}{92\!\cdots\!88}a^{19}-\frac{10\!\cdots\!67}{23\!\cdots\!22}a^{18}+\frac{92\!\cdots\!31}{23\!\cdots\!22}a^{17}-\frac{45\!\cdots\!49}{23\!\cdots\!22}a^{16}-\frac{94\!\cdots\!61}{92\!\cdots\!88}a^{15}-\frac{23\!\cdots\!67}{46\!\cdots\!44}a^{14}-\frac{89\!\cdots\!07}{92\!\cdots\!88}a^{13}-\frac{46\!\cdots\!63}{46\!\cdots\!44}a^{12}+\frac{44\!\cdots\!61}{46\!\cdots\!44}a^{11}+\frac{81\!\cdots\!65}{46\!\cdots\!44}a^{10}-\frac{43\!\cdots\!61}{92\!\cdots\!88}a^{9}-\frac{10\!\cdots\!91}{46\!\cdots\!44}a^{8}+\frac{11\!\cdots\!29}{46\!\cdots\!44}a^{7}+\frac{55\!\cdots\!59}{23\!\cdots\!22}a^{6}-\frac{95\!\cdots\!01}{46\!\cdots\!44}a^{5}+\frac{17\!\cdots\!65}{46\!\cdots\!44}a^{4}-\frac{35\!\cdots\!13}{92\!\cdots\!88}a^{3}-\frac{55\!\cdots\!99}{46\!\cdots\!44}a^{2}-\frac{24\!\cdots\!53}{92\!\cdots\!88}a-\frac{83\!\cdots\!95}{21\!\cdots\!02}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $11$ |
Class group and class number
$C_{7}\times C_{7}\times C_{273}$, which has order $13377$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $13377$ (assuming GRH)
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{733927149705626243355256479670}{230216438597189751625630235614550558131} a^{23} + \frac{6139846626389685904206661888734}{230216438597189751625630235614550558131} a^{22} + \frac{52392605393377759997657840153146}{230216438597189751625630235614550558131} a^{21} - \frac{442129578561809878982122899519900}{230216438597189751625630235614550558131} a^{20} - \frac{2070937365608864931358231622301392}{230216438597189751625630235614550558131} a^{19} + \frac{14010061499901928465736849585574332}{230216438597189751625630235614550558131} a^{18} + \frac{58295586520539610411547284158854353}{230216438597189751625630235614550558131} a^{17} - \frac{951751229309433574133757466484270975}{920865754388759006502520942458202232524} a^{16} - \frac{1150315342636605454880762748345628716}{230216438597189751625630235614550558131} a^{15} + \frac{1946384785981997987208026890404161139}{230216438597189751625630235614550558131} a^{14} + \frac{14686484537005959440113107696947958763}{230216438597189751625630235614550558131} a^{13} + \frac{1521994427551674266157873282442758595}{460432877194379503251260471229101116262} a^{12} - \frac{108091886342449971690280512895041632981}{230216438597189751625630235614550558131} a^{11} - \frac{313304173464385503137378003454708904457}{460432877194379503251260471229101116262} a^{10} + \frac{315134190574730857163656389754809118389}{230216438597189751625630235614550558131} a^{9} + \frac{4556171500138664811511983345847937553179}{920865754388759006502520942458202232524} a^{8} + \frac{791436823447647302208971663626909991366}{230216438597189751625630235614550558131} a^{7} - \frac{3572021191122057664473628012689506056535}{460432877194379503251260471229101116262} a^{6} - \frac{5035990674876727696875237801657876619194}{230216438597189751625630235614550558131} a^{5} - \frac{25063550469885657128986152097343940421169}{920865754388759006502520942458202232524} a^{4} - \frac{4626595209770945847604476409784760903498}{230216438597189751625630235614550558131} a^{3} - \frac{4048011546465336333135188592169864845871}{460432877194379503251260471229101116262} a^{2} - \frac{972438251439033402431486008207248959744}{230216438597189751625630235614550558131} a - \frac{202312286451610407618225540146386036273}{83715068580796273318410994768927475684} \)
(order $10$)
| 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: |
$\frac{36\!\cdots\!10}{23\!\cdots\!31}a^{23}-\frac{31\!\cdots\!94}{23\!\cdots\!31}a^{22}-\frac{25\!\cdots\!98}{23\!\cdots\!31}a^{21}+\frac{22\!\cdots\!98}{23\!\cdots\!31}a^{20}+\frac{10\!\cdots\!46}{23\!\cdots\!31}a^{19}-\frac{71\!\cdots\!24}{23\!\cdots\!31}a^{18}-\frac{28\!\cdots\!69}{23\!\cdots\!31}a^{17}+\frac{49\!\cdots\!49}{92\!\cdots\!24}a^{16}+\frac{56\!\cdots\!40}{23\!\cdots\!31}a^{15}-\frac{10\!\cdots\!89}{23\!\cdots\!31}a^{14}-\frac{73\!\cdots\!65}{23\!\cdots\!31}a^{13}+\frac{15\!\cdots\!77}{46\!\cdots\!62}a^{12}+\frac{55\!\cdots\!75}{23\!\cdots\!31}a^{11}+\frac{14\!\cdots\!97}{46\!\cdots\!62}a^{10}-\frac{17\!\cdots\!30}{23\!\cdots\!31}a^{9}-\frac{21\!\cdots\!71}{92\!\cdots\!24}a^{8}-\frac{28\!\cdots\!12}{23\!\cdots\!31}a^{7}+\frac{19\!\cdots\!55}{46\!\cdots\!62}a^{6}+\frac{23\!\cdots\!84}{23\!\cdots\!31}a^{5}+\frac{10\!\cdots\!29}{92\!\cdots\!24}a^{4}+\frac{19\!\cdots\!76}{23\!\cdots\!31}a^{3}+\frac{84\!\cdots\!36}{23\!\cdots\!31}a^{2}+\frac{39\!\cdots\!82}{23\!\cdots\!31}a+\frac{11\!\cdots\!49}{83\!\cdots\!84}$, $\frac{16\!\cdots\!46}{11\!\cdots\!61}a^{23}-\frac{13\!\cdots\!12}{11\!\cdots\!61}a^{22}-\frac{12\!\cdots\!84}{11\!\cdots\!61}a^{21}+\frac{10\!\cdots\!13}{11\!\cdots\!61}a^{20}+\frac{48\!\cdots\!78}{11\!\cdots\!61}a^{19}-\frac{32\!\cdots\!70}{11\!\cdots\!61}a^{18}-\frac{27\!\cdots\!05}{23\!\cdots\!22}a^{17}+\frac{22\!\cdots\!97}{46\!\cdots\!44}a^{16}+\frac{26\!\cdots\!48}{11\!\cdots\!61}a^{15}-\frac{46\!\cdots\!44}{11\!\cdots\!61}a^{14}-\frac{34\!\cdots\!13}{11\!\cdots\!61}a^{13}+\frac{16\!\cdots\!25}{23\!\cdots\!22}a^{12}+\frac{25\!\cdots\!29}{11\!\cdots\!61}a^{11}+\frac{34\!\cdots\!15}{11\!\cdots\!61}a^{10}-\frac{75\!\cdots\!72}{11\!\cdots\!61}a^{9}-\frac{10\!\cdots\!09}{46\!\cdots\!44}a^{8}-\frac{16\!\cdots\!40}{11\!\cdots\!61}a^{7}+\frac{82\!\cdots\!35}{23\!\cdots\!22}a^{6}+\frac{11\!\cdots\!88}{11\!\cdots\!61}a^{5}+\frac{56\!\cdots\!67}{46\!\cdots\!44}a^{4}+\frac{20\!\cdots\!55}{23\!\cdots\!22}a^{3}+\frac{44\!\cdots\!34}{11\!\cdots\!61}a^{2}+\frac{21\!\cdots\!30}{11\!\cdots\!61}a+\frac{51\!\cdots\!45}{42\!\cdots\!04}$, $\frac{16\!\cdots\!69}{10\!\cdots\!51}a^{23}-\frac{87\!\cdots\!45}{10\!\cdots\!51}a^{22}-\frac{13\!\cdots\!13}{84\!\cdots\!08}a^{21}+\frac{48\!\cdots\!79}{69\!\cdots\!48}a^{20}+\frac{66\!\cdots\!97}{84\!\cdots\!08}a^{19}-\frac{95\!\cdots\!03}{42\!\cdots\!04}a^{18}-\frac{19\!\cdots\!77}{84\!\cdots\!08}a^{17}+\frac{75\!\cdots\!71}{23\!\cdots\!76}a^{16}+\frac{18\!\cdots\!93}{42\!\cdots\!04}a^{15}-\frac{40\!\cdots\!45}{84\!\cdots\!08}a^{14}-\frac{55\!\cdots\!95}{10\!\cdots\!51}a^{13}-\frac{26\!\cdots\!55}{42\!\cdots\!04}a^{12}+\frac{28\!\cdots\!83}{84\!\cdots\!08}a^{11}+\frac{66\!\cdots\!37}{76\!\cdots\!28}a^{10}-\frac{51\!\cdots\!31}{84\!\cdots\!08}a^{9}-\frac{20\!\cdots\!95}{42\!\cdots\!04}a^{8}-\frac{43\!\cdots\!19}{84\!\cdots\!08}a^{7}+\frac{11\!\cdots\!27}{21\!\cdots\!02}a^{6}+\frac{88\!\cdots\!63}{42\!\cdots\!04}a^{5}+\frac{24\!\cdots\!49}{84\!\cdots\!08}a^{4}+\frac{18\!\cdots\!85}{84\!\cdots\!08}a^{3}+\frac{84\!\cdots\!19}{84\!\cdots\!08}a^{2}+\frac{51\!\cdots\!54}{10\!\cdots\!51}a+\frac{49\!\cdots\!01}{76\!\cdots\!28}$, $\frac{11\!\cdots\!91}{23\!\cdots\!22}a^{23}-\frac{58\!\cdots\!65}{92\!\cdots\!88}a^{22}-\frac{62\!\cdots\!85}{46\!\cdots\!44}a^{21}+\frac{17\!\cdots\!47}{46\!\cdots\!44}a^{20}+\frac{12\!\cdots\!13}{11\!\cdots\!61}a^{19}-\frac{52\!\cdots\!55}{46\!\cdots\!44}a^{18}-\frac{52\!\cdots\!62}{11\!\cdots\!61}a^{17}+\frac{19\!\cdots\!81}{92\!\cdots\!88}a^{16}+\frac{57\!\cdots\!39}{23\!\cdots\!22}a^{15}-\frac{59\!\cdots\!27}{23\!\cdots\!22}a^{14}-\frac{26\!\cdots\!05}{46\!\cdots\!44}a^{13}+\frac{16\!\cdots\!73}{92\!\cdots\!88}a^{12}+\frac{16\!\cdots\!91}{23\!\cdots\!22}a^{11}-\frac{25\!\cdots\!65}{11\!\cdots\!61}a^{10}-\frac{21\!\cdots\!23}{46\!\cdots\!44}a^{9}-\frac{13\!\cdots\!15}{23\!\cdots\!22}a^{8}+\frac{12\!\cdots\!09}{11\!\cdots\!61}a^{7}+\frac{37\!\cdots\!57}{92\!\cdots\!88}a^{6}+\frac{14\!\cdots\!41}{46\!\cdots\!44}a^{5}-\frac{51\!\cdots\!55}{92\!\cdots\!88}a^{4}-\frac{36\!\cdots\!15}{23\!\cdots\!22}a^{3}-\frac{15\!\cdots\!99}{92\!\cdots\!88}a^{2}-\frac{87\!\cdots\!24}{11\!\cdots\!61}a-\frac{25\!\cdots\!07}{21\!\cdots\!02}$, $\frac{25\!\cdots\!84}{11\!\cdots\!61}a^{23}-\frac{20\!\cdots\!94}{11\!\cdots\!61}a^{22}-\frac{18\!\cdots\!84}{11\!\cdots\!61}a^{21}+\frac{15\!\cdots\!21}{11\!\cdots\!61}a^{20}+\frac{73\!\cdots\!22}{11\!\cdots\!61}a^{19}-\frac{47\!\cdots\!98}{11\!\cdots\!61}a^{18}-\frac{20\!\cdots\!27}{11\!\cdots\!61}a^{17}+\frac{78\!\cdots\!51}{11\!\cdots\!61}a^{16}+\frac{40\!\cdots\!78}{11\!\cdots\!61}a^{15}-\frac{60\!\cdots\!67}{11\!\cdots\!61}a^{14}-\frac{51\!\cdots\!94}{11\!\cdots\!61}a^{13}-\frac{11\!\cdots\!11}{11\!\cdots\!61}a^{12}+\frac{36\!\cdots\!12}{11\!\cdots\!61}a^{11}+\frac{12\!\cdots\!35}{23\!\cdots\!22}a^{10}-\frac{97\!\cdots\!75}{11\!\cdots\!61}a^{9}-\frac{41\!\cdots\!41}{11\!\cdots\!61}a^{8}-\frac{34\!\cdots\!06}{11\!\cdots\!61}a^{7}+\frac{58\!\cdots\!05}{11\!\cdots\!61}a^{6}+\frac{18\!\cdots\!46}{11\!\cdots\!61}a^{5}+\frac{24\!\cdots\!11}{11\!\cdots\!61}a^{4}+\frac{18\!\cdots\!76}{11\!\cdots\!61}a^{3}+\frac{14\!\cdots\!89}{23\!\cdots\!22}a^{2}+\frac{23\!\cdots\!82}{11\!\cdots\!61}a+\frac{40\!\cdots\!79}{21\!\cdots\!02}$, $\frac{14\!\cdots\!02}{10\!\cdots\!51}a^{23}-\frac{48\!\cdots\!04}{10\!\cdots\!51}a^{22}-\frac{10\!\cdots\!66}{59\!\cdots\!69}a^{21}+\frac{21\!\cdots\!03}{42\!\cdots\!04}a^{20}+\frac{95\!\cdots\!47}{10\!\cdots\!51}a^{19}-\frac{20\!\cdots\!64}{10\!\cdots\!51}a^{18}-\frac{28\!\cdots\!21}{10\!\cdots\!51}a^{17}+\frac{12\!\cdots\!87}{42\!\cdots\!04}a^{16}+\frac{53\!\cdots\!66}{10\!\cdots\!51}a^{15}+\frac{76\!\cdots\!52}{96\!\cdots\!41}a^{14}-\frac{62\!\cdots\!25}{10\!\cdots\!51}a^{13}-\frac{13\!\cdots\!49}{21\!\cdots\!02}a^{12}+\frac{41\!\cdots\!90}{10\!\cdots\!51}a^{11}+\frac{18\!\cdots\!23}{21\!\cdots\!02}a^{10}-\frac{88\!\cdots\!96}{10\!\cdots\!51}a^{9}-\frac{20\!\cdots\!87}{42\!\cdots\!04}a^{8}-\frac{49\!\cdots\!98}{10\!\cdots\!51}a^{7}+\frac{11\!\cdots\!49}{21\!\cdots\!02}a^{6}+\frac{22\!\cdots\!10}{10\!\cdots\!51}a^{5}+\frac{54\!\cdots\!65}{19\!\cdots\!82}a^{4}+\frac{20\!\cdots\!61}{10\!\cdots\!51}a^{3}+\frac{75\!\cdots\!13}{10\!\cdots\!51}a^{2}+\frac{32\!\cdots\!50}{10\!\cdots\!51}a+\frac{21\!\cdots\!17}{96\!\cdots\!41}$, $\frac{12\!\cdots\!82}{10\!\cdots\!51}a^{23}-\frac{19\!\cdots\!69}{19\!\cdots\!82}a^{22}-\frac{84\!\cdots\!54}{10\!\cdots\!51}a^{21}+\frac{31\!\cdots\!81}{42\!\cdots\!04}a^{20}+\frac{32\!\cdots\!26}{10\!\cdots\!51}a^{19}-\frac{24\!\cdots\!08}{10\!\cdots\!51}a^{18}-\frac{89\!\cdots\!72}{10\!\cdots\!51}a^{17}+\frac{41\!\cdots\!69}{10\!\cdots\!51}a^{16}+\frac{17\!\cdots\!96}{10\!\cdots\!51}a^{15}-\frac{72\!\cdots\!57}{21\!\cdots\!02}a^{14}-\frac{21\!\cdots\!02}{96\!\cdots\!41}a^{13}+\frac{13\!\cdots\!65}{42\!\cdots\!04}a^{12}+\frac{17\!\cdots\!11}{10\!\cdots\!51}a^{11}+\frac{22\!\cdots\!35}{10\!\cdots\!51}a^{10}-\frac{55\!\cdots\!30}{10\!\cdots\!51}a^{9}-\frac{70\!\cdots\!29}{42\!\cdots\!04}a^{8}-\frac{99\!\cdots\!64}{10\!\cdots\!51}a^{7}+\frac{30\!\cdots\!76}{10\!\cdots\!51}a^{6}+\frac{76\!\cdots\!71}{10\!\cdots\!51}a^{5}+\frac{36\!\cdots\!83}{42\!\cdots\!04}a^{4}+\frac{66\!\cdots\!59}{10\!\cdots\!51}a^{3}+\frac{26\!\cdots\!02}{10\!\cdots\!51}a^{2}+\frac{79\!\cdots\!84}{10\!\cdots\!51}a+\frac{13\!\cdots\!51}{19\!\cdots\!82}$, $\frac{21\!\cdots\!53}{46\!\cdots\!44}a^{23}-\frac{21\!\cdots\!47}{46\!\cdots\!44}a^{22}-\frac{12\!\cdots\!95}{46\!\cdots\!44}a^{21}+\frac{21\!\cdots\!44}{64\!\cdots\!59}a^{20}+\frac{20\!\cdots\!25}{23\!\cdots\!22}a^{19}-\frac{49\!\cdots\!11}{46\!\cdots\!44}a^{18}-\frac{10\!\cdots\!05}{46\!\cdots\!44}a^{17}+\frac{90\!\cdots\!17}{46\!\cdots\!44}a^{16}+\frac{21\!\cdots\!71}{46\!\cdots\!44}a^{15}-\frac{48\!\cdots\!81}{23\!\cdots\!22}a^{14}-\frac{30\!\cdots\!85}{46\!\cdots\!44}a^{13}+\frac{25\!\cdots\!65}{23\!\cdots\!22}a^{12}+\frac{26\!\cdots\!73}{46\!\cdots\!44}a^{11}+\frac{36\!\cdots\!89}{46\!\cdots\!44}a^{10}-\frac{54\!\cdots\!95}{23\!\cdots\!22}a^{9}-\frac{16\!\cdots\!95}{46\!\cdots\!44}a^{8}+\frac{54\!\cdots\!39}{46\!\cdots\!44}a^{7}+\frac{12\!\cdots\!13}{11\!\cdots\!61}a^{6}+\frac{19\!\cdots\!46}{11\!\cdots\!61}a^{5}+\frac{59\!\cdots\!33}{46\!\cdots\!44}a^{4}+\frac{41\!\cdots\!47}{11\!\cdots\!61}a^{3}-\frac{99\!\cdots\!57}{11\!\cdots\!61}a^{2}+\frac{30\!\cdots\!17}{46\!\cdots\!44}a+\frac{20\!\cdots\!91}{42\!\cdots\!04}$, $\frac{33\!\cdots\!59}{46\!\cdots\!44}a^{23}-\frac{46\!\cdots\!19}{92\!\cdots\!88}a^{22}-\frac{28\!\cdots\!05}{46\!\cdots\!44}a^{21}+\frac{16\!\cdots\!13}{46\!\cdots\!44}a^{20}+\frac{12\!\cdots\!81}{46\!\cdots\!44}a^{19}-\frac{25\!\cdots\!47}{23\!\cdots\!22}a^{18}-\frac{35\!\cdots\!85}{46\!\cdots\!44}a^{17}+\frac{15\!\cdots\!07}{92\!\cdots\!88}a^{16}+\frac{17\!\cdots\!01}{11\!\cdots\!61}a^{15}-\frac{23\!\cdots\!47}{46\!\cdots\!44}a^{14}-\frac{40\!\cdots\!41}{23\!\cdots\!22}a^{13}-\frac{17\!\cdots\!05}{92\!\cdots\!88}a^{12}+\frac{49\!\cdots\!89}{46\!\cdots\!44}a^{11}+\frac{67\!\cdots\!91}{23\!\cdots\!22}a^{10}-\frac{55\!\cdots\!75}{46\!\cdots\!44}a^{9}-\frac{35\!\cdots\!61}{23\!\cdots\!22}a^{8}-\frac{51\!\cdots\!17}{23\!\cdots\!22}a^{7}+\frac{66\!\cdots\!69}{92\!\cdots\!88}a^{6}+\frac{17\!\cdots\!89}{23\!\cdots\!22}a^{5}+\frac{12\!\cdots\!87}{92\!\cdots\!88}a^{4}+\frac{29\!\cdots\!53}{23\!\cdots\!22}a^{3}+\frac{64\!\cdots\!59}{92\!\cdots\!88}a^{2}+\frac{19\!\cdots\!65}{11\!\cdots\!61}a-\frac{11\!\cdots\!25}{21\!\cdots\!02}$, $\frac{55\!\cdots\!83}{92\!\cdots\!88}a^{23}-\frac{11\!\cdots\!29}{23\!\cdots\!22}a^{22}-\frac{39\!\cdots\!91}{92\!\cdots\!88}a^{21}+\frac{84\!\cdots\!07}{23\!\cdots\!22}a^{20}+\frac{15\!\cdots\!05}{92\!\cdots\!88}a^{19}-\frac{53\!\cdots\!47}{46\!\cdots\!44}a^{18}-\frac{21\!\cdots\!61}{46\!\cdots\!44}a^{17}+\frac{50\!\cdots\!63}{25\!\cdots\!36}a^{16}+\frac{41\!\cdots\!89}{46\!\cdots\!44}a^{15}-\frac{75\!\cdots\!57}{46\!\cdots\!44}a^{14}-\frac{10\!\cdots\!15}{92\!\cdots\!88}a^{13}+\frac{57\!\cdots\!25}{46\!\cdots\!44}a^{12}+\frac{76\!\cdots\!09}{92\!\cdots\!88}a^{11}+\frac{26\!\cdots\!31}{23\!\cdots\!22}a^{10}-\frac{21\!\cdots\!85}{92\!\cdots\!88}a^{9}-\frac{95\!\cdots\!20}{11\!\cdots\!61}a^{8}-\frac{78\!\cdots\!02}{11\!\cdots\!61}a^{7}+\frac{44\!\cdots\!33}{46\!\cdots\!44}a^{6}+\frac{32\!\cdots\!29}{92\!\cdots\!88}a^{5}+\frac{26\!\cdots\!81}{46\!\cdots\!44}a^{4}+\frac{14\!\cdots\!69}{23\!\cdots\!22}a^{3}+\frac{83\!\cdots\!99}{23\!\cdots\!22}a^{2}+\frac{35\!\cdots\!61}{46\!\cdots\!44}a-\frac{15\!\cdots\!57}{10\!\cdots\!51}$, $\frac{28\!\cdots\!63}{92\!\cdots\!88}a^{23}-\frac{36\!\cdots\!59}{92\!\cdots\!88}a^{22}-\frac{87\!\cdots\!65}{92\!\cdots\!88}a^{21}+\frac{24\!\cdots\!51}{92\!\cdots\!88}a^{20}-\frac{50\!\cdots\!89}{46\!\cdots\!44}a^{19}-\frac{38\!\cdots\!31}{46\!\cdots\!44}a^{18}+\frac{32\!\cdots\!59}{92\!\cdots\!88}a^{17}+\frac{15\!\cdots\!21}{92\!\cdots\!88}a^{16}-\frac{20\!\cdots\!37}{92\!\cdots\!88}a^{15}-\frac{19\!\cdots\!73}{92\!\cdots\!88}a^{14}-\frac{87\!\cdots\!87}{92\!\cdots\!88}a^{13}+\frac{16\!\cdots\!25}{92\!\cdots\!88}a^{12}+\frac{19\!\cdots\!21}{92\!\cdots\!88}a^{11}-\frac{82\!\cdots\!67}{92\!\cdots\!88}a^{10}-\frac{22\!\cdots\!59}{11\!\cdots\!61}a^{9}+\frac{67\!\cdots\!91}{46\!\cdots\!44}a^{8}+\frac{73\!\cdots\!23}{92\!\cdots\!88}a^{7}+\frac{59\!\cdots\!25}{92\!\cdots\!88}a^{6}-\frac{18\!\cdots\!91}{23\!\cdots\!22}a^{5}-\frac{10\!\cdots\!77}{46\!\cdots\!44}a^{4}-\frac{11\!\cdots\!09}{46\!\cdots\!44}a^{3}-\frac{14\!\cdots\!64}{11\!\cdots\!61}a^{2}-\frac{39\!\cdots\!45}{92\!\cdots\!88}a-\frac{37\!\cdots\!61}{84\!\cdots\!08}$
| 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: | \( 398411333053.5562 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 398411333053.5562 \cdot 13377}{10\cdot\sqrt{3234204723240544858872018632704000000000000000000}}\cr\approx \mathstrut & 1.12192823428927 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 - 74*x^22 + 572*x^21 + 3016*x^20 - 17760*x^19 - 85518*x^18 + 285704*x^17 + 1661305*x^16 - 1906632*x^15 - 20467124*x^14 - 10151352*x^13 + 139734456*x^12 + 275550256*x^11 - 290937638*x^10 - 1685333160*x^9 - 1899102168*x^8 + 1612921144*x^7 + 7900602944*x^6 + 12266775908*x^5 + 11114510270*x^4 + 6111392440*x^3 + 2347621720*x^2 + 1082467988*x + 466788641)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^24 - 8*x^23 - 74*x^22 + 572*x^21 + 3016*x^20 - 17760*x^19 - 85518*x^18 + 285704*x^17 + 1661305*x^16 - 1906632*x^15 - 20467124*x^14 - 10151352*x^13 + 139734456*x^12 + 275550256*x^11 - 290937638*x^10 - 1685333160*x^9 - 1899102168*x^8 + 1612921144*x^7 + 7900602944*x^6 + 12266775908*x^5 + 11114510270*x^4 + 6111392440*x^3 + 2347621720*x^2 + 1082467988*x + 466788641, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 8*x^23 - 74*x^22 + 572*x^21 + 3016*x^20 - 17760*x^19 - 85518*x^18 + 285704*x^17 + 1661305*x^16 - 1906632*x^15 - 20467124*x^14 - 10151352*x^13 + 139734456*x^12 + 275550256*x^11 - 290937638*x^10 - 1685333160*x^9 - 1899102168*x^8 + 1612921144*x^7 + 7900602944*x^6 + 12266775908*x^5 + 11114510270*x^4 + 6111392440*x^3 + 2347621720*x^2 + 1082467988*x + 466788641);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 8*x^23 - 74*x^22 + 572*x^21 + 3016*x^20 - 17760*x^19 - 85518*x^18 + 285704*x^17 + 1661305*x^16 - 1906632*x^15 - 20467124*x^14 - 10151352*x^13 + 139734456*x^12 + 275550256*x^11 - 290937638*x^10 - 1685333160*x^9 - 1899102168*x^8 + 1612921144*x^7 + 7900602944*x^6 + 12266775908*x^5 + 11114510270*x^4 + 6111392440*x^3 + 2347621720*x^2 + 1082467988*x + 466788641);
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
$C_2\times C_{12}$ (as 24T2):
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)
| An abelian group of order 24 |
| The 24 conjugacy class representatives for $C_2\times C_{12}$ |
| Character table for $C_2\times C_{12}$ |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | R | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.1.0.1}{1} }^{24}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{6}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $24$ | $2$ | $12$ | $36$ | |||
|
\(5\)
| Deg $24$ | $4$ | $6$ | $18$ | |||
|
\(37\)
| 37.12.8.1 | $x^{12} + 18 x^{10} + 220 x^{9} + 114 x^{8} + 864 x^{7} - 5754 x^{6} + 7320 x^{5} - 47346 x^{4} - 240044 x^{3} + 340080 x^{2} - 2045220 x + 8612757$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
| 37.12.8.1 | $x^{12} + 18 x^{10} + 220 x^{9} + 114 x^{8} + 864 x^{7} - 5754 x^{6} + 7320 x^{5} - 47346 x^{4} - 240044 x^{3} + 340080 x^{2} - 2045220 x + 8612757$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |