Normalized defining polynomial
\( x^{12} + 364 x^{10} + 31213 x^{8} + 979524 x^{6} + 12073698 x^{4} + 48294792 x^{2} + 54331641 \)
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(476278906078322265274202456064=2^{24}\cdot 3^{6}\cdot 7^{10}\cdot 13^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $297.27$ | 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: | $2, 3, 7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2184=2^{3}\cdot 3\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2184}(1537,·)$, $\chi_{2184}(1091,·)$, $\chi_{2184}(901,·)$, $\chi_{2184}(1739,·)$, $\chi_{2184}(1,·)$, $\chi_{2184}(263,·)$, $\chi_{2184}(911,·)$, $\chi_{2184}(1811,·)$, $\chi_{2184}(181,·)$, $\chi_{2184}(1465,·)$, $\chi_{2184}(829,·)$, $\chi_{2184}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{819} a^{6} + \frac{4}{9} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{2457} a^{7} + \frac{4}{27} a^{5} - \frac{8}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{243243} a^{8} + \frac{58}{243243} a^{6} + \frac{91}{2673} a^{4} - \frac{98}{297} a^{2} + \frac{4}{33}$, $\frac{1}{729729} a^{9} + \frac{58}{729729} a^{7} + \frac{91}{8019} a^{5} - \frac{395}{891} a^{3} + \frac{37}{99} a$, $\frac{1}{77319895653} a^{10} - \frac{20525}{77319895653} a^{8} - \frac{14652107}{77319895653} a^{6} - \frac{1043000}{10489743} a^{4} - \frac{1708049}{3496581} a^{2} + \frac{461597}{1165527}$, $\frac{1}{231959686959} a^{11} - \frac{20525}{231959686959} a^{9} - \frac{14652107}{231959686959} a^{7} - \frac{1043000}{31469229} a^{5} - \frac{5204630}{10489743} a^{3} - \frac{703930}{3496581} a$
Class group and class number
$C_{2}\times C_{2}\times C_{12}\times C_{36}\times C_{684}$, which has order $1181952$ (assuming GRH)
Unit group
| Rank: | $5$ | 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: | \( 131497.04163948935 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_6$ (as 12T2):
| An abelian group of order 12 |
| The 12 conjugacy class representatives for $C_6\times C_2$ |
| Character table for $C_6\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-182}) \), \(\Q(\sqrt{-546}) \), \(\Q(\sqrt{3}) \), 3.3.8281.2, \(\Q(\sqrt{3}, \sqrt{-182})\), 6.0.3195044582912.4, 6.0.86266203738624.2, 6.6.118497532608.2 |
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 | R | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.24.307 | $x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $7$ | 7.12.10.2 | $x^{12} + 35 x^{6} + 441$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
| $13$ | 13.6.5.2 | $x^{6} - 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.2 | $x^{6} - 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |