Normalized defining polynomial
\( x^{12} - x^{11} - 20 x^{10} + 9 x^{9} + 51 x^{8} + 470 x^{7} - 1456 x^{6} - 2454 x^{5} + 22006 x^{4} - 53748 x^{3} + 64152 x^{2} - 36036 x + 6268 \)
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(14116592251679730040832=2^{25}\cdot 29^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $70.11$ | 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: | $2, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{334} a^{10} + \frac{13}{334} a^{9} + \frac{15}{167} a^{8} + \frac{49}{334} a^{7} + \frac{117}{334} a^{6} - \frac{9}{167} a^{5} - \frac{59}{167} a^{4} - \frac{30}{167} a^{3} + \frac{1}{167} a^{2} - \frac{21}{167} a - \frac{80}{167}$, $\frac{1}{79290639144363478} a^{11} - \frac{26465367060571}{39645319572181739} a^{10} - \frac{944124871520679}{79290639144363478} a^{9} - \frac{3784270038159305}{79290639144363478} a^{8} - \frac{14165062449755219}{39645319572181739} a^{7} + \frac{2907617706818803}{6099279934181806} a^{6} + \frac{1421023793534382}{3049639967090903} a^{5} + \frac{5774767358434160}{39645319572181739} a^{4} - \frac{13789721162155280}{39645319572181739} a^{3} + \frac{4082694892008546}{39645319572181739} a^{2} + \frac{13335190401500119}{39645319572181739} a + \frac{7351698396855031}{39645319572181739}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 3242345.9302 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $A_4:C_4$ |
| Character table for $A_4:C_4$ |
Intermediate fields
| 3.3.6728.1, 6.2.1448511488.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | 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 | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | R | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.2 | $x^{2} + 6$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.8.22.7 | $x^{8} + 2 x^{4} + 16 x + 4$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ | |
| $29$ | 29.12.10.3 | $x^{12} + 232 x^{6} + 22707$ | $6$ | $2$ | $10$ | $C_3 : C_4$ | $[\ ]_{6}^{2}$ |