Normalized defining polynomial
\( x^{12} - 2 x^{11} - 102 x^{10} + 199 x^{9} + 3833 x^{8} - 6937 x^{7} - 66192 x^{6} + 105589 x^{5} + 514762 x^{4} - 647862 x^{3} - 1393261 x^{2} + 752713 x + 494507 \)
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(16551311845247868759889=7^{8}\cdot 13^{6}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $71.05$ | 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: | $7, 13, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{808205216138548434674288} a^{11} - \frac{92258070733775557289937}{808205216138548434674288} a^{10} + \frac{180757463625116396774121}{808205216138548434674288} a^{9} + \frac{10397260370695558018135}{101025652017318554334286} a^{8} - \frac{89988756435395265526919}{808205216138548434674288} a^{7} + \frac{21506235533601848812260}{50512826008659277167143} a^{6} + \frac{24628035600212751377714}{50512826008659277167143} a^{5} - \frac{304903063582356431434539}{808205216138548434674288} a^{4} - \frac{286808739373228999851969}{808205216138548434674288} a^{3} - \frac{246608638943645487887975}{808205216138548434674288} a^{2} - \frac{69124158396550804885171}{202051304034637108668572} a - \frac{83293928929037492205195}{808205216138548434674288}$
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 2120183.06022 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times A_4$ (as 12T7):
| A solvable group of order 24 |
| The 8 conjugacy class representatives for $A_4 \times C_2$ |
| Character table for $A_4 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{377}) \), \(\Q(\zeta_{7})^+\), 6.6.905177.1, 6.6.128651901833.1, 6.6.341251729.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 6 sibling: | data not computed |
| Degree 8 sibling: | data not computed |
| Degree 12 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | R | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $29$ | 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |