Normalized defining polynomial
\( x^{24} - x^{23} - 37 x^{22} + 36 x^{21} + 551 x^{20} - 515 x^{19} - 4277 x^{18} + 3762 x^{17} + 18936 x^{16} - 15174 x^{15} - 50091 x^{14} + 34917 x^{13} + 81925 x^{12} - 46487 x^{11} - 84021 x^{10} + 34947 x^{9} + 53243 x^{8} - 13356 x^{7} - 19591 x^{6} + 1711 x^{5} + 3541 x^{4} + 172 x^{3} - 180 x^{2} - 12 x + 1 \)
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[24, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(416724332309376501938461264338134765625=3^{12}\cdot 5^{12}\cdot 13^{22}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.66$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(195=3\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(11,·)$, $\chi_{195}(4,·)$, $\chi_{195}(71,·)$, $\chi_{195}(139,·)$, $\chi_{195}(79,·)$, $\chi_{195}(16,·)$, $\chi_{195}(149,·)$, $\chi_{195}(86,·)$, $\chi_{195}(89,·)$, $\chi_{195}(94,·)$, $\chi_{195}(161,·)$, $\chi_{195}(164,·)$, $\chi_{195}(166,·)$, $\chi_{195}(41,·)$, $\chi_{195}(44,·)$, $\chi_{195}(176,·)$, $\chi_{195}(49,·)$, $\chi_{195}(181,·)$, $\chi_{195}(119,·)$, $\chi_{195}(121,·)$, $\chi_{195}(59,·)$, $\chi_{195}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{79} a^{21} - \frac{6}{79} a^{20} - \frac{32}{79} a^{19} - \frac{14}{79} a^{18} + \frac{26}{79} a^{17} + \frac{7}{79} a^{16} - \frac{1}{79} a^{15} - \frac{1}{79} a^{14} + \frac{14}{79} a^{13} - \frac{7}{79} a^{12} - \frac{39}{79} a^{11} - \frac{10}{79} a^{10} - \frac{8}{79} a^{9} - \frac{4}{79} a^{8} - \frac{16}{79} a^{7} + \frac{8}{79} a^{6} - \frac{20}{79} a^{5} - \frac{33}{79} a^{4} - \frac{2}{79} a^{3} + \frac{29}{79} a^{2} + \frac{9}{79}$, $\frac{1}{79} a^{22} + \frac{11}{79} a^{20} + \frac{31}{79} a^{19} + \frac{21}{79} a^{18} + \frac{5}{79} a^{17} - \frac{38}{79} a^{16} - \frac{7}{79} a^{15} + \frac{8}{79} a^{14} - \frac{2}{79} a^{13} - \frac{2}{79} a^{12} - \frac{7}{79} a^{11} + \frac{11}{79} a^{10} + \frac{27}{79} a^{9} + \frac{39}{79} a^{8} - \frac{9}{79} a^{7} + \frac{28}{79} a^{6} + \frac{5}{79} a^{5} + \frac{37}{79} a^{4} + \frac{17}{79} a^{3} + \frac{16}{79} a^{2} + \frac{9}{79} a - \frac{25}{79}$, $\frac{1}{13460641917638295063947537} a^{23} + \frac{20646348816497656155988}{13460641917638295063947537} a^{22} + \frac{8889193458833009151937}{13460641917638295063947537} a^{21} + \frac{5786642062920334141316583}{13460641917638295063947537} a^{20} - \frac{107756855684354444857788}{253974375804496133282029} a^{19} + \frac{5640475635450590784053870}{13460641917638295063947537} a^{18} - \frac{6404365649507561270316431}{13460641917638295063947537} a^{17} + \frac{1595642365223327210359748}{13460641917638295063947537} a^{16} + \frac{207032020716775086794309}{13460641917638295063947537} a^{15} + \frac{1474352310127376343555613}{13460641917638295063947537} a^{14} - \frac{67572179969070894754697}{253974375804496133282029} a^{13} - \frac{14183632664908395351571}{34603192590329807362333} a^{12} - \frac{3690112599666086890366905}{13460641917638295063947537} a^{11} + \frac{523569968835307609570118}{13460641917638295063947537} a^{10} + \frac{2342430744206253647323575}{13460641917638295063947537} a^{9} + \frac{885717411599243525104291}{13460641917638295063947537} a^{8} - \frac{5357584092632817025638274}{13460641917638295063947537} a^{7} + \frac{5764808345900927230121792}{13460641917638295063947537} a^{6} + \frac{2933704594881301379591467}{13460641917638295063947537} a^{5} + \frac{5516059352536196609025201}{13460641917638295063947537} a^{4} + \frac{1031278658942323842586715}{13460641917638295063947537} a^{3} + \frac{1162356660211931731270697}{13460641917638295063947537} a^{2} - \frac{6490353566159683344886421}{13460641917638295063947537} a - \frac{5824605622886034144242912}{13460641917638295063947537}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $23$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 340731139253.48956 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
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 $C_2\times C_{12}$ is not computed |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | R | R | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 13 | Data not computed | ||||||