Normalized defining polynomial
\( x^{12} - 6 x^{11} + 61 x^{10} - 250 x^{9} + 1244 x^{8} - 3542 x^{7} + 8985 x^{6} - 15542 x^{5} + 32704 x^{4} - 43170 x^{3} + 56065 x^{2} - 36550 x + 84475 \)
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: | \(564668382613504000000=2^{18}\cdot 5^{6}\cdot 13^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $53.62$ | 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: | $2, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(520=2^{3}\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{520}(1,·)$, $\chi_{520}(259,·)$, $\chi_{520}(51,·)$, $\chi_{520}(289,·)$, $\chi_{520}(9,·)$, $\chi_{520}(209,·)$, $\chi_{520}(491,·)$, $\chi_{520}(321,·)$, $\chi_{520}(81,·)$, $\chi_{520}(179,·)$, $\chi_{520}(251,·)$, $\chi_{520}(459,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{20} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} + \frac{3}{20} a^{4} + \frac{3}{10} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{20} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{4} a^{5} - \frac{1}{10} a^{4} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{572998900} a^{10} - \frac{1}{114599780} a^{9} - \frac{2008717}{286499450} a^{8} + \frac{8034883}{286499450} a^{7} + \frac{19825173}{114599780} a^{6} - \frac{67122347}{572998900} a^{5} - \frac{40348911}{286499450} a^{4} - \frac{44992487}{286499450} a^{3} - \frac{17743131}{114599780} a^{2} - \frac{14231369}{114599780} a - \frac{3669549}{11459978}$, $\frac{1}{10142653528900} a^{11} + \frac{1769}{2028530705780} a^{10} + \frac{97033302031}{10142653528900} a^{9} - \frac{7052903859}{327182371900} a^{8} - \frac{402855880993}{2028530705780} a^{7} + \frac{2348657958103}{10142653528900} a^{6} - \frac{145254780457}{10142653528900} a^{5} + \frac{1834232847911}{10142653528900} a^{4} - \frac{390209703811}{2028530705780} a^{3} - \frac{368546294687}{2028530705780} a^{2} + \frac{55005793805}{405706141156} a - \frac{3397749993}{13087294876}$
Class group and class number
$C_{2}\times C_{2}\times C_{252}$, which has order $1008$ (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: | \( 615.5445050404002 \) (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
| \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-130}) \), \(\Q(\sqrt{5}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{-26})\), 6.0.190102016.1, 6.0.23762752000.2, 6.6.3570125.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.18.15 | $x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $13$ | 13.12.10.1 | $x^{12} - 117 x^{6} + 10816$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |