Normalized defining polynomial
\( x^{12} - 2 x^{11} + 9 x^{10} - 14 x^{9} + 287 x^{8} - 366 x^{7} + 4117 x^{6} - 3108 x^{5} + 39521 x^{4} - 23428 x^{3} + 264195 x^{2} - 99100 x + 697801 \)
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: | \(2548231104297008664576=2^{12}\cdot 3^{6}\cdot 7^{8}\cdot 23^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.79$ | 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: | $2, 3, 7, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1932=2^{2}\cdot 3\cdot 7\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1932}(1,·)$, $\chi_{1932}(323,·)$, $\chi_{1932}(1381,·)$, $\chi_{1932}(1703,·)$, $\chi_{1932}(781,·)$, $\chi_{1932}(1103,·)$, $\chi_{1932}(275,·)$, $\chi_{1932}(277,·)$, $\chi_{1932}(599,·)$, $\chi_{1932}(505,·)$, $\chi_{1932}(827,·)$, $\chi_{1932}(1885,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{35} a^{9} + \frac{16}{35} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{7} a^{5} - \frac{3}{35} a^{4} - \frac{1}{7} a^{3} - \frac{16}{35} a^{2} + \frac{9}{35} a + \frac{13}{35}$, $\frac{1}{1275959755} a^{10} + \frac{11571661}{1275959755} a^{9} - \frac{16268152}{1275959755} a^{8} + \frac{64960747}{182279965} a^{7} - \frac{60084618}{255191951} a^{6} + \frac{218685672}{1275959755} a^{5} + \frac{31117580}{255191951} a^{4} + \frac{17984262}{182279965} a^{3} + \frac{12888981}{26039995} a^{2} + \frac{329586608}{1275959755} a + \frac{56719674}{255191951}$, $\frac{1}{64924661489624755} a^{11} - \frac{9605733}{64924661489624755} a^{10} + \frac{263953912099922}{64924661489624755} a^{9} - \frac{62597811910803}{142691563713461} a^{8} + \frac{27479798632592983}{64924661489624755} a^{7} + \frac{15064011316517949}{64924661489624755} a^{6} - \frac{25136143540027528}{64924661489624755} a^{5} - \frac{54537242923608}{142691563713461} a^{4} - \frac{6290658872236}{9274951641374965} a^{3} - \frac{4181687638819711}{64924661489624755} a^{2} - \frac{1738500650506840}{12984932297924951} a + \frac{12880554031262}{101922545509615}$
Class group and class number
$C_{6}\times C_{234}$, which has order $1404$ (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: | \( 553.0667020684372 \) (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
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{-23}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{3}, \sqrt{-23})\), 6.6.4148928.1, 6.0.50480006976.3, 6.0.29212967.1 |
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.6.5 | $x^{6} - 2 x^{4} + x^{2} - 3$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.6.6.5 | $x^{6} - 2 x^{4} + x^{2} - 3$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| $3$ | 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $7$ | 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| $23$ | 23.6.3.2 | $x^{6} - 529 x^{2} + 48668$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 23.6.3.2 | $x^{6} - 529 x^{2} + 48668$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |