Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 38*x^22 + 118*x^21 + 410*x^20 - 4159*x^19 - 6137*x^18 + 66684*x^17 + 364189*x^16 - 617183*x^15 - 5509477*x^14 + 20246396*x^13 + 30856912*x^12 - 480510060*x^11 - 262994586*x^10 + 1469803588*x^9 - 1179057951*x^8 + 5895919061*x^7 + 61423569727*x^6 + 164020859052*x^5 + 248952353934*x^4 + 251235790407*x^3 + 376139557176*x^2 + 30591970887*x + 113554323611)
gp: K = bnfinit(y^24 - y^23 - 38*y^22 + 118*y^21 + 410*y^20 - 4159*y^19 - 6137*y^18 + 66684*y^17 + 364189*y^16 - 617183*y^15 - 5509477*y^14 + 20246396*y^13 + 30856912*y^12 - 480510060*y^11 - 262994586*y^10 + 1469803588*y^9 - 1179057951*y^8 + 5895919061*y^7 + 61423569727*y^6 + 164020859052*y^5 + 248952353934*y^4 + 251235790407*y^3 + 376139557176*y^2 + 30591970887*y + 113554323611, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 38*x^22 + 118*x^21 + 410*x^20 - 4159*x^19 - 6137*x^18 + 66684*x^17 + 364189*x^16 - 617183*x^15 - 5509477*x^14 + 20246396*x^13 + 30856912*x^12 - 480510060*x^11 - 262994586*x^10 + 1469803588*x^9 - 1179057951*x^8 + 5895919061*x^7 + 61423569727*x^6 + 164020859052*x^5 + 248952353934*x^4 + 251235790407*x^3 + 376139557176*x^2 + 30591970887*x + 113554323611);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - x^23 - 38*x^22 + 118*x^21 + 410*x^20 - 4159*x^19 - 6137*x^18 + 66684*x^17 + 364189*x^16 - 617183*x^15 - 5509477*x^14 + 20246396*x^13 + 30856912*x^12 - 480510060*x^11 - 262994586*x^10 + 1469803588*x^9 - 1179057951*x^8 + 5895919061*x^7 + 61423569727*x^6 + 164020859052*x^5 + 248952353934*x^4 + 251235790407*x^3 + 376139557176*x^2 + 30591970887*x + 113554323611)
\( x^{24} - x^{23} - 38 x^{22} + 118 x^{21} + 410 x^{20} - 4159 x^{19} - 6137 x^{18} + 66684 x^{17} + \cdots + 113554323611 \)
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: |
\(4280568617115764108693321069663602825199127197265625\)
\(\medspace = 5^{18}\cdot 7^{20}\cdot 17^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(141.68\) | 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: | $5^{3/4}7^{5/6}17^{3/4}\approx 141.681309097028$ | ||
| Ramified primes: |
\(5\), \(7\), \(17\)
| 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: | \(595=5\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{595}(256,·)$, $\chi_{595}(1,·)$, $\chi_{595}(579,·)$, $\chi_{595}(132,·)$, $\chi_{595}(69,·)$, $\chi_{595}(582,·)$, $\chi_{595}(72,·)$, $\chi_{595}(268,·)$, $\chi_{595}(271,·)$, $\chi_{595}(208,·)$, $\chi_{595}(86,·)$, $\chi_{595}(472,·)$, $\chi_{595}(409,·)$, $\chi_{595}(101,·)$, $\chi_{595}(356,·)$, $\chi_{595}(293,·)$, $\chi_{595}(38,·)$, $\chi_{595}(424,·)$, $\chi_{595}(169,·)$, $\chi_{595}(47,·)$, $\chi_{595}(242,·)$, $\chi_{595}(438,·)$, $\chi_{595}(183,·)$, $\chi_{595}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{59}a^{21}-\frac{21}{59}a^{20}+\frac{2}{59}a^{19}+\frac{10}{59}a^{18}+\frac{13}{59}a^{17}-\frac{7}{59}a^{16}-\frac{26}{59}a^{15}-\frac{9}{59}a^{14}+\frac{3}{59}a^{13}-\frac{12}{59}a^{12}+\frac{26}{59}a^{11}-\frac{29}{59}a^{10}-\frac{14}{59}a^{9}-\frac{21}{59}a^{8}-\frac{16}{59}a^{7}-\frac{24}{59}a^{6}-\frac{20}{59}a^{5}+\frac{24}{59}a^{4}+\frac{26}{59}a^{3}-\frac{3}{59}a^{2}+\frac{24}{59}a+\frac{19}{59}$, $\frac{1}{32\!\cdots\!73}a^{22}-\frac{308399670642}{32\!\cdots\!73}a^{21}+\frac{192620802716535}{32\!\cdots\!73}a^{20}-\frac{12\!\cdots\!71}{32\!\cdots\!73}a^{19}-\frac{645891004472952}{32\!\cdots\!73}a^{18}+\frac{595100471565461}{32\!\cdots\!73}a^{17}-\frac{148805198980098}{32\!\cdots\!73}a^{16}+\frac{803950255544627}{32\!\cdots\!73}a^{15}-\frac{11\!\cdots\!38}{32\!\cdots\!73}a^{14}+\frac{10\!\cdots\!84}{32\!\cdots\!73}a^{13}+\frac{11\!\cdots\!02}{32\!\cdots\!73}a^{12}+\frac{790983417829922}{32\!\cdots\!73}a^{11}+\frac{12\!\cdots\!58}{32\!\cdots\!73}a^{10}+\frac{976828692287524}{32\!\cdots\!73}a^{9}-\frac{676394963281004}{32\!\cdots\!73}a^{8}+\frac{165537969107968}{32\!\cdots\!73}a^{7}+\frac{287162738978819}{32\!\cdots\!73}a^{6}+\frac{926739370902129}{32\!\cdots\!73}a^{5}+\frac{11\!\cdots\!52}{32\!\cdots\!73}a^{4}-\frac{556735441895296}{32\!\cdots\!73}a^{3}+\frac{12\!\cdots\!84}{32\!\cdots\!73}a^{2}-\frac{15\!\cdots\!89}{32\!\cdots\!73}a+\frac{11\!\cdots\!57}{32\!\cdots\!73}$, $\frac{1}{14\!\cdots\!37}a^{23}+\frac{19\!\cdots\!33}{14\!\cdots\!37}a^{22}+\frac{44\!\cdots\!64}{14\!\cdots\!37}a^{21}+\frac{65\!\cdots\!27}{14\!\cdots\!37}a^{20}+\frac{60\!\cdots\!43}{14\!\cdots\!37}a^{19}-\frac{29\!\cdots\!02}{14\!\cdots\!37}a^{18}+\frac{59\!\cdots\!51}{14\!\cdots\!37}a^{17}+\frac{12\!\cdots\!13}{14\!\cdots\!37}a^{16}-\frac{61\!\cdots\!96}{14\!\cdots\!37}a^{15}-\frac{99\!\cdots\!85}{14\!\cdots\!37}a^{14}-\frac{13\!\cdots\!72}{14\!\cdots\!37}a^{13}+\frac{27\!\cdots\!42}{14\!\cdots\!37}a^{12}-\frac{84\!\cdots\!42}{14\!\cdots\!37}a^{11}-\frac{14\!\cdots\!17}{14\!\cdots\!37}a^{10}+\frac{21\!\cdots\!90}{14\!\cdots\!37}a^{9}+\frac{55\!\cdots\!58}{14\!\cdots\!37}a^{8}-\frac{44\!\cdots\!91}{14\!\cdots\!37}a^{7}+\frac{32\!\cdots\!69}{14\!\cdots\!37}a^{6}+\frac{64\!\cdots\!79}{14\!\cdots\!37}a^{5}-\frac{57\!\cdots\!79}{14\!\cdots\!37}a^{4}+\frac{15\!\cdots\!09}{14\!\cdots\!37}a^{3}-\frac{69\!\cdots\!83}{14\!\cdots\!37}a^{2}+\frac{20\!\cdots\!64}{14\!\cdots\!37}a-\frac{22\!\cdots\!41}{14\!\cdots\!37}$
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
$C_{2}\times C_{14920}$, which has order $29840$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $7460$ (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: |
\( -1 \)
(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: |
$\frac{47\!\cdots\!86}{70\!\cdots\!91}a^{23}-\frac{95\!\cdots\!75}{70\!\cdots\!91}a^{22}-\frac{18\!\cdots\!02}{70\!\cdots\!91}a^{21}+\frac{75\!\cdots\!36}{70\!\cdots\!91}a^{20}+\frac{16\!\cdots\!74}{70\!\cdots\!91}a^{19}-\frac{22\!\cdots\!58}{70\!\cdots\!91}a^{18}-\frac{11\!\cdots\!92}{70\!\cdots\!91}a^{17}+\frac{37\!\cdots\!11}{70\!\cdots\!91}a^{16}+\frac{14\!\cdots\!16}{70\!\cdots\!91}a^{15}-\frac{50\!\cdots\!71}{70\!\cdots\!91}a^{14}-\frac{25\!\cdots\!37}{70\!\cdots\!91}a^{13}+\frac{12\!\cdots\!87}{70\!\cdots\!91}a^{12}+\frac{82\!\cdots\!69}{70\!\cdots\!91}a^{11}-\frac{25\!\cdots\!27}{70\!\cdots\!91}a^{10}+\frac{85\!\cdots\!72}{70\!\cdots\!91}a^{9}+\frac{11\!\cdots\!20}{70\!\cdots\!91}a^{8}-\frac{12\!\cdots\!27}{70\!\cdots\!91}a^{7}+\frac{20\!\cdots\!24}{70\!\cdots\!91}a^{6}+\frac{27\!\cdots\!73}{70\!\cdots\!91}a^{5}+\frac{47\!\cdots\!61}{70\!\cdots\!91}a^{4}+\frac{11\!\cdots\!98}{70\!\cdots\!91}a^{3}-\frac{66\!\cdots\!11}{72\!\cdots\!03}a^{2}+\frac{16\!\cdots\!93}{70\!\cdots\!91}a-\frac{84\!\cdots\!21}{70\!\cdots\!91}$, $\frac{73\!\cdots\!05}{70\!\cdots\!91}a^{23}-\frac{22\!\cdots\!85}{70\!\cdots\!91}a^{22}-\frac{33\!\cdots\!65}{70\!\cdots\!91}a^{21}+\frac{73\!\cdots\!35}{70\!\cdots\!91}a^{20}+\frac{56\!\cdots\!40}{70\!\cdots\!91}a^{19}-\frac{35\!\cdots\!40}{70\!\cdots\!91}a^{18}-\frac{88\!\cdots\!50}{70\!\cdots\!91}a^{17}+\frac{69\!\cdots\!10}{70\!\cdots\!91}a^{16}+\frac{32\!\cdots\!61}{70\!\cdots\!91}a^{15}-\frac{66\!\cdots\!05}{70\!\cdots\!91}a^{14}-\frac{61\!\cdots\!10}{70\!\cdots\!91}a^{13}+\frac{16\!\cdots\!25}{70\!\cdots\!91}a^{12}+\frac{63\!\cdots\!25}{70\!\cdots\!91}a^{11}-\frac{45\!\cdots\!50}{70\!\cdots\!91}a^{10}-\frac{61\!\cdots\!00}{70\!\cdots\!91}a^{9}+\frac{36\!\cdots\!95}{70\!\cdots\!91}a^{8}+\frac{72\!\cdots\!85}{70\!\cdots\!91}a^{7}-\frac{88\!\cdots\!85}{70\!\cdots\!91}a^{6}+\frac{56\!\cdots\!37}{70\!\cdots\!91}a^{5}+\frac{14\!\cdots\!70}{70\!\cdots\!91}a^{4}-\frac{67\!\cdots\!55}{70\!\cdots\!91}a^{3}-\frac{44\!\cdots\!40}{72\!\cdots\!03}a^{2}+\frac{54\!\cdots\!50}{70\!\cdots\!91}a+\frac{13\!\cdots\!80}{70\!\cdots\!91}$, $\frac{44\!\cdots\!29}{14\!\cdots\!37}a^{23}-\frac{23\!\cdots\!18}{14\!\cdots\!37}a^{22}-\frac{14\!\cdots\!78}{14\!\cdots\!37}a^{21}+\frac{36\!\cdots\!95}{14\!\cdots\!37}a^{20}+\frac{11\!\cdots\!91}{14\!\cdots\!37}a^{19}-\frac{12\!\cdots\!17}{14\!\cdots\!37}a^{18}-\frac{33\!\cdots\!61}{14\!\cdots\!37}a^{17}+\frac{15\!\cdots\!89}{14\!\cdots\!37}a^{16}+\frac{18\!\cdots\!96}{14\!\cdots\!37}a^{15}-\frac{12\!\cdots\!16}{14\!\cdots\!37}a^{14}-\frac{20\!\cdots\!63}{14\!\cdots\!37}a^{13}+\frac{44\!\cdots\!20}{14\!\cdots\!37}a^{12}+\frac{91\!\cdots\!90}{14\!\cdots\!37}a^{11}-\frac{12\!\cdots\!21}{14\!\cdots\!37}a^{10}-\frac{28\!\cdots\!47}{14\!\cdots\!37}a^{9}-\frac{70\!\cdots\!26}{14\!\cdots\!37}a^{8}+\frac{22\!\cdots\!79}{14\!\cdots\!37}a^{7}+\frac{54\!\cdots\!66}{14\!\cdots\!37}a^{6}+\frac{13\!\cdots\!59}{14\!\cdots\!37}a^{5}+\frac{12\!\cdots\!52}{14\!\cdots\!37}a^{4}+\frac{27\!\cdots\!76}{14\!\cdots\!37}a^{3}+\frac{92\!\cdots\!38}{14\!\cdots\!37}a^{2}+\frac{11\!\cdots\!41}{14\!\cdots\!37}a+\frac{30\!\cdots\!17}{14\!\cdots\!37}$, $\frac{47\!\cdots\!53}{14\!\cdots\!37}a^{23}-\frac{23\!\cdots\!97}{14\!\cdots\!37}a^{22}-\frac{16\!\cdots\!00}{14\!\cdots\!37}a^{21}+\frac{13\!\cdots\!27}{14\!\cdots\!37}a^{20}-\frac{48\!\cdots\!39}{14\!\cdots\!37}a^{19}-\frac{30\!\cdots\!95}{14\!\cdots\!37}a^{18}+\frac{64\!\cdots\!49}{14\!\cdots\!37}a^{17}+\frac{45\!\cdots\!82}{14\!\cdots\!37}a^{16}+\frac{10\!\cdots\!95}{14\!\cdots\!37}a^{15}-\frac{99\!\cdots\!64}{14\!\cdots\!37}a^{14}-\frac{90\!\cdots\!19}{14\!\cdots\!37}a^{13}+\frac{22\!\cdots\!94}{14\!\cdots\!37}a^{12}-\frac{32\!\cdots\!76}{14\!\cdots\!37}a^{11}-\frac{32\!\cdots\!52}{14\!\cdots\!37}a^{10}+\frac{10\!\cdots\!92}{14\!\cdots\!37}a^{9}+\frac{12\!\cdots\!03}{14\!\cdots\!37}a^{8}-\frac{73\!\cdots\!86}{14\!\cdots\!37}a^{7}+\frac{75\!\cdots\!15}{14\!\cdots\!37}a^{6}+\frac{26\!\cdots\!77}{14\!\cdots\!37}a^{5}-\frac{41\!\cdots\!10}{14\!\cdots\!37}a^{4}-\frac{15\!\cdots\!48}{14\!\cdots\!37}a^{3}+\frac{75\!\cdots\!28}{14\!\cdots\!37}a^{2}+\frac{62\!\cdots\!44}{14\!\cdots\!37}a-\frac{32\!\cdots\!29}{14\!\cdots\!37}$, $\frac{35\!\cdots\!18}{22\!\cdots\!63}a^{23}-\frac{44\!\cdots\!39}{22\!\cdots\!63}a^{22}-\frac{14\!\cdots\!32}{22\!\cdots\!63}a^{21}+\frac{46\!\cdots\!52}{22\!\cdots\!63}a^{20}+\frac{18\!\cdots\!95}{22\!\cdots\!63}a^{19}-\frac{16\!\cdots\!11}{22\!\cdots\!63}a^{18}-\frac{24\!\cdots\!07}{22\!\cdots\!63}a^{17}+\frac{30\!\cdots\!66}{22\!\cdots\!63}a^{16}+\frac{13\!\cdots\!76}{22\!\cdots\!63}a^{15}-\frac{35\!\cdots\!43}{22\!\cdots\!63}a^{14}-\frac{23\!\cdots\!97}{22\!\cdots\!63}a^{13}+\frac{87\!\cdots\!08}{22\!\cdots\!63}a^{12}+\frac{17\!\cdots\!27}{22\!\cdots\!63}a^{11}-\frac{20\!\cdots\!41}{22\!\cdots\!63}a^{10}-\frac{10\!\cdots\!96}{22\!\cdots\!63}a^{9}+\frac{12\!\cdots\!55}{22\!\cdots\!63}a^{8}-\frac{22\!\cdots\!94}{22\!\cdots\!63}a^{7}-\frac{12\!\cdots\!29}{22\!\cdots\!63}a^{6}+\frac{23\!\cdots\!72}{22\!\cdots\!63}a^{5}+\frac{51\!\cdots\!16}{22\!\cdots\!63}a^{4}-\frac{12\!\cdots\!92}{22\!\cdots\!63}a^{3}-\frac{12\!\cdots\!60}{22\!\cdots\!63}a^{2}+\frac{18\!\cdots\!99}{22\!\cdots\!63}a+\frac{10\!\cdots\!47}{22\!\cdots\!63}$, $\frac{54\!\cdots\!80}{22\!\cdots\!63}a^{23}-\frac{11\!\cdots\!35}{22\!\cdots\!63}a^{22}-\frac{20\!\cdots\!54}{22\!\cdots\!63}a^{21}+\frac{90\!\cdots\!73}{22\!\cdots\!63}a^{20}+\frac{15\!\cdots\!98}{22\!\cdots\!63}a^{19}-\frac{26\!\cdots\!50}{22\!\cdots\!63}a^{18}-\frac{55\!\cdots\!76}{22\!\cdots\!63}a^{17}+\frac{42\!\cdots\!31}{22\!\cdots\!63}a^{16}+\frac{15\!\cdots\!93}{22\!\cdots\!63}a^{15}-\frac{59\!\cdots\!34}{22\!\cdots\!63}a^{14}-\frac{26\!\cdots\!90}{22\!\cdots\!63}a^{13}+\frac{15\!\cdots\!94}{22\!\cdots\!63}a^{12}+\frac{45\!\cdots\!94}{22\!\cdots\!63}a^{11}-\frac{29\!\cdots\!90}{22\!\cdots\!63}a^{10}+\frac{19\!\cdots\!39}{22\!\cdots\!63}a^{9}+\frac{11\!\cdots\!55}{22\!\cdots\!63}a^{8}-\frac{21\!\cdots\!41}{22\!\cdots\!63}a^{7}+\frac{43\!\cdots\!03}{22\!\cdots\!63}a^{6}+\frac{30\!\cdots\!47}{22\!\cdots\!63}a^{5}+\frac{40\!\cdots\!28}{22\!\cdots\!63}a^{4}+\frac{29\!\cdots\!82}{22\!\cdots\!63}a^{3}-\frac{38\!\cdots\!64}{22\!\cdots\!63}a^{2}+\frac{16\!\cdots\!63}{22\!\cdots\!63}a-\frac{20\!\cdots\!93}{22\!\cdots\!63}$, $\frac{24\!\cdots\!33}{12\!\cdots\!23}a^{23}-\frac{50\!\cdots\!51}{12\!\cdots\!23}a^{22}-\frac{94\!\cdots\!38}{12\!\cdots\!23}a^{21}+\frac{39\!\cdots\!50}{12\!\cdots\!23}a^{20}+\frac{81\!\cdots\!22}{12\!\cdots\!23}a^{19}-\frac{11\!\cdots\!01}{12\!\cdots\!23}a^{18}-\frac{61\!\cdots\!71}{12\!\cdots\!23}a^{17}+\frac{19\!\cdots\!69}{12\!\cdots\!23}a^{16}+\frac{76\!\cdots\!17}{12\!\cdots\!23}a^{15}-\frac{26\!\cdots\!36}{12\!\cdots\!23}a^{14}-\frac{13\!\cdots\!59}{12\!\cdots\!23}a^{13}+\frac{65\!\cdots\!57}{12\!\cdots\!23}a^{12}+\frac{40\!\cdots\!87}{12\!\cdots\!23}a^{11}-\frac{13\!\cdots\!12}{12\!\cdots\!23}a^{10}+\frac{41\!\cdots\!05}{12\!\cdots\!23}a^{9}+\frac{55\!\cdots\!30}{12\!\cdots\!23}a^{8}-\frac{51\!\cdots\!05}{12\!\cdots\!23}a^{7}+\frac{93\!\cdots\!23}{12\!\cdots\!23}a^{6}+\frac{13\!\cdots\!64}{12\!\cdots\!23}a^{5}+\frac{26\!\cdots\!08}{12\!\cdots\!23}a^{4}+\frac{38\!\cdots\!23}{12\!\cdots\!23}a^{3}-\frac{33\!\cdots\!32}{12\!\cdots\!23}a^{2}+\frac{78\!\cdots\!95}{12\!\cdots\!23}a-\frac{16\!\cdots\!28}{12\!\cdots\!23}$, $\frac{23\!\cdots\!51}{14\!\cdots\!37}a^{23}+\frac{48\!\cdots\!67}{14\!\cdots\!37}a^{22}-\frac{96\!\cdots\!95}{14\!\cdots\!37}a^{21}+\frac{17\!\cdots\!59}{14\!\cdots\!37}a^{20}+\frac{14\!\cdots\!90}{14\!\cdots\!37}a^{19}-\frac{92\!\cdots\!50}{14\!\cdots\!37}a^{18}-\frac{27\!\cdots\!95}{14\!\cdots\!37}a^{17}+\frac{16\!\cdots\!59}{14\!\cdots\!37}a^{16}+\frac{10\!\cdots\!24}{14\!\cdots\!37}a^{15}-\frac{78\!\cdots\!16}{14\!\cdots\!37}a^{14}-\frac{16\!\cdots\!60}{14\!\cdots\!37}a^{13}+\frac{36\!\cdots\!15}{14\!\cdots\!37}a^{12}+\frac{15\!\cdots\!12}{14\!\cdots\!37}a^{11}-\frac{11\!\cdots\!20}{14\!\cdots\!37}a^{10}-\frac{34\!\cdots\!24}{24\!\cdots\!43}a^{9}+\frac{53\!\cdots\!69}{14\!\cdots\!37}a^{8}+\frac{50\!\cdots\!77}{14\!\cdots\!37}a^{7}+\frac{18\!\cdots\!94}{14\!\cdots\!37}a^{6}+\frac{18\!\cdots\!63}{14\!\cdots\!37}a^{5}+\frac{49\!\cdots\!07}{14\!\cdots\!37}a^{4}+\frac{72\!\cdots\!97}{14\!\cdots\!37}a^{3}+\frac{80\!\cdots\!06}{14\!\cdots\!37}a^{2}+\frac{56\!\cdots\!19}{14\!\cdots\!37}a+\frac{22\!\cdots\!46}{14\!\cdots\!37}$, $\frac{11\!\cdots\!82}{14\!\cdots\!37}a^{23}-\frac{71\!\cdots\!31}{14\!\cdots\!37}a^{22}-\frac{43\!\cdots\!45}{14\!\cdots\!37}a^{21}+\frac{11\!\cdots\!62}{14\!\cdots\!37}a^{20}+\frac{52\!\cdots\!01}{14\!\cdots\!37}a^{19}-\frac{45\!\cdots\!36}{14\!\cdots\!37}a^{18}-\frac{88\!\cdots\!24}{14\!\cdots\!37}a^{17}+\frac{74\!\cdots\!57}{14\!\cdots\!37}a^{16}+\frac{44\!\cdots\!78}{14\!\cdots\!37}a^{15}-\frac{56\!\cdots\!21}{14\!\cdots\!37}a^{14}-\frac{66\!\cdots\!13}{14\!\cdots\!37}a^{13}+\frac{20\!\cdots\!93}{14\!\cdots\!37}a^{12}+\frac{45\!\cdots\!86}{14\!\cdots\!37}a^{11}-\frac{53\!\cdots\!85}{14\!\cdots\!37}a^{10}-\frac{51\!\cdots\!10}{14\!\cdots\!37}a^{9}+\frac{17\!\cdots\!57}{14\!\cdots\!37}a^{8}-\frac{54\!\cdots\!24}{14\!\cdots\!37}a^{7}+\frac{56\!\cdots\!14}{14\!\cdots\!37}a^{6}+\frac{72\!\cdots\!46}{14\!\cdots\!37}a^{5}+\frac{21\!\cdots\!55}{14\!\cdots\!37}a^{4}+\frac{32\!\cdots\!06}{14\!\cdots\!37}a^{3}+\frac{32\!\cdots\!51}{14\!\cdots\!37}a^{2}+\frac{42\!\cdots\!06}{14\!\cdots\!37}a+\frac{77\!\cdots\!29}{14\!\cdots\!37}$, $\frac{53\!\cdots\!19}{14\!\cdots\!37}a^{23}+\frac{60\!\cdots\!29}{14\!\cdots\!37}a^{22}-\frac{21\!\cdots\!37}{14\!\cdots\!37}a^{21}+\frac{40\!\cdots\!57}{14\!\cdots\!37}a^{20}+\frac{30\!\cdots\!80}{14\!\cdots\!37}a^{19}-\frac{20\!\cdots\!39}{14\!\cdots\!37}a^{18}-\frac{60\!\cdots\!18}{14\!\cdots\!37}a^{17}+\frac{33\!\cdots\!95}{14\!\cdots\!37}a^{16}+\frac{24\!\cdots\!05}{14\!\cdots\!37}a^{15}-\frac{13\!\cdots\!04}{14\!\cdots\!37}a^{14}-\frac{35\!\cdots\!78}{14\!\cdots\!37}a^{13}+\frac{75\!\cdots\!29}{14\!\cdots\!37}a^{12}+\frac{31\!\cdots\!35}{14\!\cdots\!37}a^{11}-\frac{24\!\cdots\!09}{14\!\cdots\!37}a^{10}-\frac{46\!\cdots\!01}{14\!\cdots\!37}a^{9}+\frac{78\!\cdots\!11}{14\!\cdots\!37}a^{8}+\frac{83\!\cdots\!31}{14\!\cdots\!37}a^{7}+\frac{20\!\cdots\!87}{14\!\cdots\!37}a^{6}+\frac{34\!\cdots\!43}{14\!\cdots\!37}a^{5}+\frac{12\!\cdots\!16}{14\!\cdots\!37}a^{4}+\frac{20\!\cdots\!56}{14\!\cdots\!37}a^{3}+\frac{16\!\cdots\!53}{14\!\cdots\!37}a^{2}+\frac{12\!\cdots\!82}{14\!\cdots\!37}a+\frac{48\!\cdots\!29}{14\!\cdots\!37}$, $\frac{89\!\cdots\!95}{19\!\cdots\!07}a^{23}+\frac{52\!\cdots\!04}{19\!\cdots\!31}a^{22}-\frac{71\!\cdots\!86}{19\!\cdots\!07}a^{21}-\frac{35\!\cdots\!44}{19\!\cdots\!07}a^{20}+\frac{21\!\cdots\!71}{19\!\cdots\!07}a^{19}-\frac{69\!\cdots\!98}{19\!\cdots\!07}a^{18}-\frac{34\!\cdots\!40}{19\!\cdots\!07}a^{17}+\frac{16\!\cdots\!01}{19\!\cdots\!07}a^{16}+\frac{67\!\cdots\!09}{19\!\cdots\!07}a^{15}-\frac{47\!\cdots\!67}{19\!\cdots\!07}a^{14}-\frac{16\!\cdots\!68}{19\!\cdots\!07}a^{13}+\frac{18\!\cdots\!91}{19\!\cdots\!07}a^{12}+\frac{27\!\cdots\!58}{19\!\cdots\!07}a^{11}-\frac{11\!\cdots\!49}{19\!\cdots\!07}a^{10}-\frac{27\!\cdots\!99}{19\!\cdots\!31}a^{9}+\frac{14\!\cdots\!55}{19\!\cdots\!07}a^{8}-\frac{96\!\cdots\!75}{19\!\cdots\!07}a^{7}-\frac{21\!\cdots\!32}{19\!\cdots\!07}a^{6}+\frac{17\!\cdots\!43}{19\!\cdots\!07}a^{5}+\frac{14\!\cdots\!22}{19\!\cdots\!07}a^{4}-\frac{37\!\cdots\!09}{19\!\cdots\!07}a^{3}-\frac{16\!\cdots\!56}{19\!\cdots\!07}a^{2}+\frac{24\!\cdots\!60}{19\!\cdots\!07}a-\frac{32\!\cdots\!31}{19\!\cdots\!07}$
| 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: | \( 166017367069.493 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 166017367069.493 \cdot 29840}{2\cdot\sqrt{4280568617115764108693321069663602825199127197265625}}\cr\approx \mathstrut & 0.143327667678980 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 38*x^22 + 118*x^21 + 410*x^20 - 4159*x^19 - 6137*x^18 + 66684*x^17 + 364189*x^16 - 617183*x^15 - 5509477*x^14 + 20246396*x^13 + 30856912*x^12 - 480510060*x^11 - 262994586*x^10 + 1469803588*x^9 - 1179057951*x^8 + 5895919061*x^7 + 61423569727*x^6 + 164020859052*x^5 + 248952353934*x^4 + 251235790407*x^3 + 376139557176*x^2 + 30591970887*x + 113554323611)
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 - x^23 - 38*x^22 + 118*x^21 + 410*x^20 - 4159*x^19 - 6137*x^18 + 66684*x^17 + 364189*x^16 - 617183*x^15 - 5509477*x^14 + 20246396*x^13 + 30856912*x^12 - 480510060*x^11 - 262994586*x^10 + 1469803588*x^9 - 1179057951*x^8 + 5895919061*x^7 + 61423569727*x^6 + 164020859052*x^5 + 248952353934*x^4 + 251235790407*x^3 + 376139557176*x^2 + 30591970887*x + 113554323611, 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 - x^23 - 38*x^22 + 118*x^21 + 410*x^20 - 4159*x^19 - 6137*x^18 + 66684*x^17 + 364189*x^16 - 617183*x^15 - 5509477*x^14 + 20246396*x^13 + 30856912*x^12 - 480510060*x^11 - 262994586*x^10 + 1469803588*x^9 - 1179057951*x^8 + 5895919061*x^7 + 61423569727*x^6 + 164020859052*x^5 + 248952353934*x^4 + 251235790407*x^3 + 376139557176*x^2 + 30591970887*x + 113554323611);
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 - x^23 - 38*x^22 + 118*x^21 + 410*x^20 - 4159*x^19 - 6137*x^18 + 66684*x^17 + 364189*x^16 - 617183*x^15 - 5509477*x^14 + 20246396*x^13 + 30856912*x^12 - 480510060*x^11 - 262994586*x^10 + 1469803588*x^9 - 1179057951*x^8 + 5895919061*x^7 + 61423569727*x^6 + 164020859052*x^5 + 248952353934*x^4 + 251235790407*x^3 + 376139557176*x^2 + 30591970887*x + 113554323611);
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 | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{4}$ | R | R | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.3 | $x^{12} + 75 x^{4} - 375$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.3 | $x^{12} + 75 x^{4} - 375$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
|
\(7\)
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
|
\(17\)
| 17.12.9.3 | $x^{12} + 289 x^{4} - 68782$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 17.12.9.3 | $x^{12} + 289 x^{4} - 68782$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |