Normalized defining polynomial
\( x^{24} - x^{23} - 20 x^{22} + 20 x^{21} + 320 x^{20} - 219 x^{19} - 4901 x^{18} + 14901 x^{17} + \cdots + 1061520150601 \)
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: |
\(177340502563276529989135238501590728759765625\)
\(\medspace = 5^{18}\cdot 7^{20}\cdot 17^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.78\) | 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: | $5^{3/4}7^{5/6}17^{1/2}\approx 69.77507799528695$ | ||
| 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$$\chi_{595}(256,·)$, $\chi_{595}(1,·)$, $\chi_{595}(67,·)$, $\chi_{595}(324,·)$, $\chi_{595}(69,·)$, $\chi_{595}(577,·)$, $\chi_{595}(458,·)$, $\chi_{595}(579,·)$, $\chi_{595}(341,·)$, $\chi_{595}(86,·)$, $\chi_{595}(407,·)$, $\chi_{595}(152,·)$, $\chi_{595}(409,·)$, $\chi_{595}(543,·)$, $\chi_{595}(288,·)$, $\chi_{595}(33,·)$, $\chi_{595}(426,·)$, $\chi_{595}(171,·)$, $\chi_{595}(492,·)$, $\chi_{595}(237,·)$, $\chi_{595}(494,·)$, $\chi_{595}(239,·)$, $\chi_{595}(373,·)$, $\chi_{595}(118,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29}a^{18}-\frac{5}{29}a^{17}+\frac{13}{29}a^{16}+\frac{13}{29}a^{15}+\frac{8}{29}a^{14}+\frac{9}{29}a^{13}+\frac{6}{29}a^{12}-\frac{6}{29}a^{11}+\frac{4}{29}a^{10}-\frac{14}{29}a^{9}+\frac{1}{29}a^{8}+\frac{3}{29}a^{7}+\frac{11}{29}a^{6}+\frac{14}{29}a^{5}-\frac{4}{29}a^{4}-\frac{8}{29}a^{3}-\frac{8}{29}a^{2}+\frac{6}{29}a+\frac{9}{29}$, $\frac{1}{57\!\cdots\!49}a^{19}+\frac{51\!\cdots\!74}{57\!\cdots\!49}a^{18}-\frac{13\!\cdots\!84}{57\!\cdots\!49}a^{17}+\frac{29\!\cdots\!09}{57\!\cdots\!49}a^{16}-\frac{13\!\cdots\!45}{57\!\cdots\!49}a^{15}+\frac{15\!\cdots\!99}{57\!\cdots\!49}a^{14}-\frac{15\!\cdots\!31}{57\!\cdots\!49}a^{13}-\frac{13\!\cdots\!94}{57\!\cdots\!49}a^{12}+\frac{26\!\cdots\!65}{57\!\cdots\!49}a^{11}+\frac{27\!\cdots\!27}{57\!\cdots\!49}a^{10}+\frac{17\!\cdots\!35}{57\!\cdots\!49}a^{9}-\frac{20\!\cdots\!69}{57\!\cdots\!49}a^{8}+\frac{14\!\cdots\!89}{57\!\cdots\!49}a^{7}+\frac{17\!\cdots\!01}{57\!\cdots\!49}a^{6}+\frac{24\!\cdots\!78}{57\!\cdots\!49}a^{5}-\frac{26\!\cdots\!62}{57\!\cdots\!49}a^{4}+\frac{34\!\cdots\!31}{57\!\cdots\!49}a^{3}-\frac{69\!\cdots\!43}{19\!\cdots\!81}a^{2}+\frac{20\!\cdots\!35}{57\!\cdots\!49}a-\frac{34\!\cdots\!85}{57\!\cdots\!49}$, $\frac{1}{58\!\cdots\!49}a^{20}-\frac{1}{58\!\cdots\!49}a^{19}+\frac{36\!\cdots\!24}{58\!\cdots\!49}a^{18}-\frac{58\!\cdots\!17}{20\!\cdots\!81}a^{17}-\frac{58\!\cdots\!91}{58\!\cdots\!49}a^{16}-\frac{24\!\cdots\!08}{58\!\cdots\!49}a^{15}-\frac{28\!\cdots\!99}{58\!\cdots\!49}a^{14}+\frac{24\!\cdots\!48}{58\!\cdots\!49}a^{13}-\frac{70\!\cdots\!39}{20\!\cdots\!81}a^{12}-\frac{10\!\cdots\!95}{14\!\cdots\!89}a^{11}-\frac{22\!\cdots\!98}{58\!\cdots\!49}a^{10}-\frac{74\!\cdots\!36}{58\!\cdots\!49}a^{9}+\frac{83\!\cdots\!15}{58\!\cdots\!49}a^{8}+\frac{84\!\cdots\!12}{58\!\cdots\!49}a^{7}-\frac{12\!\cdots\!46}{58\!\cdots\!49}a^{6}+\frac{60\!\cdots\!72}{58\!\cdots\!49}a^{5}+\frac{25\!\cdots\!18}{58\!\cdots\!49}a^{4}-\frac{19\!\cdots\!49}{58\!\cdots\!49}a^{3}-\frac{26\!\cdots\!34}{58\!\cdots\!49}a^{2}-\frac{27\!\cdots\!99}{57\!\cdots\!49}a+\frac{94\!\cdots\!96}{57\!\cdots\!49}$, $\frac{1}{59\!\cdots\!49}a^{21}-\frac{1}{59\!\cdots\!49}a^{20}-\frac{20}{59\!\cdots\!49}a^{19}+\frac{11\!\cdots\!12}{59\!\cdots\!49}a^{18}-\frac{14\!\cdots\!70}{59\!\cdots\!49}a^{17}+\frac{12\!\cdots\!39}{59\!\cdots\!49}a^{16}-\frac{44\!\cdots\!46}{59\!\cdots\!49}a^{15}-\frac{22\!\cdots\!89}{59\!\cdots\!49}a^{14}-\frac{28\!\cdots\!21}{59\!\cdots\!49}a^{13}+\frac{25\!\cdots\!56}{59\!\cdots\!49}a^{12}+\frac{17\!\cdots\!89}{59\!\cdots\!49}a^{11}+\frac{31\!\cdots\!74}{59\!\cdots\!49}a^{10}-\frac{16\!\cdots\!15}{59\!\cdots\!49}a^{9}+\frac{26\!\cdots\!84}{59\!\cdots\!49}a^{8}+\frac{25\!\cdots\!07}{59\!\cdots\!49}a^{7}+\frac{23\!\cdots\!60}{59\!\cdots\!49}a^{6}+\frac{38\!\cdots\!03}{59\!\cdots\!49}a^{5}-\frac{24\!\cdots\!83}{59\!\cdots\!49}a^{4}-\frac{94\!\cdots\!79}{59\!\cdots\!49}a^{3}-\frac{26\!\cdots\!96}{20\!\cdots\!81}a^{2}-\frac{17\!\cdots\!11}{57\!\cdots\!49}a+\frac{15\!\cdots\!49}{57\!\cdots\!49}$, $\frac{1}{59\!\cdots\!49}a^{22}-\frac{1}{59\!\cdots\!49}a^{21}-\frac{20}{59\!\cdots\!49}a^{20}+\frac{20}{59\!\cdots\!49}a^{19}+\frac{10\!\cdots\!77}{59\!\cdots\!49}a^{18}-\frac{44\!\cdots\!05}{59\!\cdots\!49}a^{17}-\frac{17\!\cdots\!12}{59\!\cdots\!49}a^{16}-\frac{58\!\cdots\!83}{59\!\cdots\!49}a^{15}-\frac{38\!\cdots\!31}{59\!\cdots\!49}a^{14}-\frac{16\!\cdots\!53}{59\!\cdots\!49}a^{13}+\frac{29\!\cdots\!91}{59\!\cdots\!49}a^{12}+\frac{16\!\cdots\!13}{14\!\cdots\!89}a^{11}-\frac{18\!\cdots\!92}{59\!\cdots\!49}a^{10}-\frac{18\!\cdots\!94}{59\!\cdots\!49}a^{9}-\frac{15\!\cdots\!86}{59\!\cdots\!49}a^{8}-\frac{23\!\cdots\!39}{59\!\cdots\!49}a^{7}-\frac{21\!\cdots\!45}{59\!\cdots\!49}a^{6}+\frac{80\!\cdots\!59}{59\!\cdots\!49}a^{5}-\frac{88\!\cdots\!70}{59\!\cdots\!49}a^{4}+\frac{11\!\cdots\!13}{59\!\cdots\!49}a^{3}+\frac{34\!\cdots\!50}{58\!\cdots\!49}a^{2}-\frac{90\!\cdots\!92}{57\!\cdots\!49}a-\frac{79\!\cdots\!51}{57\!\cdots\!49}$, $\frac{1}{60\!\cdots\!49}a^{23}-\frac{1}{60\!\cdots\!49}a^{22}-\frac{20}{60\!\cdots\!49}a^{21}+\frac{20}{60\!\cdots\!49}a^{20}+\frac{320}{60\!\cdots\!49}a^{19}-\frac{47\!\cdots\!38}{60\!\cdots\!49}a^{18}-\frac{27\!\cdots\!04}{60\!\cdots\!49}a^{17}-\frac{68\!\cdots\!62}{14\!\cdots\!89}a^{16}+\frac{23\!\cdots\!14}{60\!\cdots\!49}a^{15}-\frac{37\!\cdots\!90}{60\!\cdots\!49}a^{14}-\frac{24\!\cdots\!19}{60\!\cdots\!49}a^{13}-\frac{26\!\cdots\!52}{60\!\cdots\!49}a^{12}-\frac{95\!\cdots\!88}{60\!\cdots\!49}a^{11}-\frac{21\!\cdots\!51}{60\!\cdots\!49}a^{10}+\frac{38\!\cdots\!50}{60\!\cdots\!49}a^{9}+\frac{25\!\cdots\!26}{60\!\cdots\!49}a^{8}-\frac{25\!\cdots\!95}{60\!\cdots\!49}a^{7}-\frac{12\!\cdots\!24}{60\!\cdots\!49}a^{6}-\frac{19\!\cdots\!83}{60\!\cdots\!49}a^{5}-\frac{89\!\cdots\!67}{59\!\cdots\!49}a^{4}+\frac{19\!\cdots\!09}{59\!\cdots\!49}a^{3}+\frac{16\!\cdots\!53}{58\!\cdots\!49}a^{2}+\frac{52\!\cdots\!76}{57\!\cdots\!49}a+\frac{27\!\cdots\!91}{57\!\cdots\!49}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Relative class number: data not computed
Unit group
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{74349846400}{19951503126289682510881} a^{22} + \frac{1121512435864619}{19951503126289682510881} a^{15} + \frac{370601601649420740}{486622027470480061241} a^{8} - \frac{5994729475730497208240}{19951503126289682510881} a \)
(order $14$)
| 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: | not computed | 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: | not computed | 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 = \mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot R \cdot h}{14\cdot\sqrt{177340502563276529989135238501590728759765625}}\cr\mathstrut & \text{
Galois group
$C_2\times C_{12}$ (as 24T2):
| 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.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{24}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $24$ | $4$ | $6$ | $18$ | |||
|
\(7\)
| Deg $24$ | $6$ | $4$ | $20$ | |||
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |