Normalized defining polynomial
\( x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 294 x^{8} - 514 x^{7} + 4907 x^{6} - 8316 x^{5} + 50219 x^{4} - 58470 x^{3} + 274854 x^{2} - 156080 x + 680545 \)
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: | \(2435758881214464000000=2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.56$ | 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: | $2, 3, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1560=2^{3}\cdot 3\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1560}(1,·)$, $\chi_{1560}(581,·)$, $\chi_{1560}(289,·)$, $\chi_{1560}(841,·)$, $\chi_{1560}(1249,·)$, $\chi_{1560}(269,·)$, $\chi_{1560}(29,·)$, $\chi_{1560}(529,·)$, $\chi_{1560}(1301,·)$, $\chi_{1560}(601,·)$, $\chi_{1560}(989,·)$, $\chi_{1560}(341,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{145} a^{9} + \frac{13}{145} a^{8} + \frac{12}{29} a^{7} - \frac{33}{145} a^{6} + \frac{17}{145} a^{5} + \frac{3}{29} a^{4} + \frac{33}{145} a^{3} + \frac{16}{145} a^{2} - \frac{4}{29} a + \frac{5}{29}$, $\frac{1}{144616475} a^{10} - \frac{23483}{28923295} a^{9} + \frac{6037421}{144616475} a^{8} - \frac{22756028}{144616475} a^{7} - \frac{59014549}{144616475} a^{6} - \frac{46342626}{144616475} a^{5} - \frac{32222407}{144616475} a^{4} + \frac{46311567}{144616475} a^{3} + \frac{69117917}{144616475} a^{2} - \frac{5752726}{28923295} a - \frac{2317048}{28923295}$, $\frac{1}{10964787728094275} a^{11} + \frac{1000929}{353702829938525} a^{10} + \frac{28412104517156}{10964787728094275} a^{9} - \frac{220599084091094}{10964787728094275} a^{8} + \frac{2993523314705719}{10964787728094275} a^{7} + \frac{2814604391036468}{10964787728094275} a^{6} - \frac{1786681098511486}{10964787728094275} a^{5} + \frac{1466163156650559}{10964787728094275} a^{4} + \frac{78958659487323}{438591509123771} a^{3} - \frac{3192054375277442}{10964787728094275} a^{2} + \frac{147137955198843}{2192957545618855} a - \frac{484155922950022}{2192957545618855}$
Class group and class number
$C_{2}\times C_{6}\times C_{156}$, which has order $1872$ (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
| \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{-6})\), 6.0.49353408000.11, 6.6.3570125.1, 6.0.394827264.1 |
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 | R | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.18.23 | $x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
| $3$ | 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 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 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $13$ | 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |