Normalized defining polynomial
\( x^{12} - 4 x^{11} + 79 x^{10} - 248 x^{9} + 2800 x^{8} - 6806 x^{7} + 56388 x^{6} - 101293 x^{5} + 679716 x^{4} - 811595 x^{3} + 4660873 x^{2} - 2792379 x + 14297009 \)
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: | \(3816011369976660765625=5^{6}\cdot 11^{6}\cdot 13^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $62.87$ | 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: | $5, 11, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(715=5\cdot 11\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(386,·)$, $\chi_{715}(549,·)$, $\chi_{715}(166,·)$, $\chi_{715}(329,·)$, $\chi_{715}(714,·)$, $\chi_{715}(274,·)$, $\chi_{715}(659,·)$, $\chi_{715}(276,·)$, $\chi_{715}(439,·)$, $\chi_{715}(56,·)$, $\chi_{715}(441,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{384195066434528277176784953} a^{11} - \frac{91703896015889832599683204}{384195066434528277176784953} a^{10} - \frac{111927225146819987788124001}{384195066434528277176784953} a^{9} + \frac{33979557460934107201262772}{384195066434528277176784953} a^{8} + \frac{98201264463847114607221063}{384195066434528277176784953} a^{7} - \frac{102365223610876749903058141}{384195066434528277176784953} a^{6} + \frac{50904065494621580239340009}{384195066434528277176784953} a^{5} + \frac{139579428233594821818334088}{384195066434528277176784953} a^{4} + \frac{75853026282122425113992344}{384195066434528277176784953} a^{3} - \frac{74552114245291785893007798}{384195066434528277176784953} a^{2} - \frac{105752436823178329305924908}{384195066434528277176784953} a - \frac{1052975300518702953665460}{2122624676433857884954613}$
Class group and class number
$C_{4}\times C_{4}\times C_{168}$, which has order $2688$ (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: | \( 120.78403136265631 \) (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
| \(\Q(\sqrt{-715}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-55})\), 6.0.61773872875.3, 6.0.4751836375.3, \(\Q(\zeta_{13})^+\) |
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 | ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ | R | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | 11.12.6.1 | $x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| $13$ | 13.6.5.2 | $x^{6} - 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.2 | $x^{6} - 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |