Normalized defining polynomial
\( x^{12} + 22 x^{8} - 55 x^{4} + 176 \)
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: | \(18698185645162496=2^{16}\cdot 11^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.70$ | 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: | $2, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{7} - \frac{1}{4} a^{5} + \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{144} a^{8} - \frac{7}{48} a^{4} - \frac{1}{2} a^{2} - \frac{1}{9}$, $\frac{1}{288} a^{9} - \frac{1}{288} a^{8} + \frac{17}{96} a^{5} - \frac{17}{96} a^{4} - \frac{1}{18} a + \frac{1}{18}$, $\frac{1}{288} a^{10} - \frac{1}{288} a^{8} + \frac{5}{96} a^{6} - \frac{17}{96} a^{4} + \frac{5}{72} a^{2} - \frac{4}{9}$, $\frac{1}{576} a^{11} - \frac{1}{576} a^{10} - \frac{1}{576} a^{9} + \frac{1}{576} a^{8} + \frac{5}{192} a^{7} + \frac{7}{192} a^{6} + \frac{31}{192} a^{5} + \frac{17}{192} a^{4} - \frac{31}{144} a^{3} + \frac{29}{72} a^{2} - \frac{2}{9} a - \frac{1}{36}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5788.39813773 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 24 |
| The 9 conjugacy class representatives for $D_{12}$ |
| Character table for $D_{12}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 3.1.484.1, 4.0.21296.1, 6.0.2576816.1 |
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 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 | R | ${\href{/LocalNumberField/3.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/13.12.0.1}{12} }$ | ${\href{/LocalNumberField/17.12.0.1}{12} }$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.4 | $x^{4} - 5$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.8.12.14 | $x^{8} + 12 x^{4} + 144$ | $4$ | $2$ | $12$ | $D_4$ | $[2, 2]^{2}$ | |
| $11$ | 11.12.11.1 | $x^{12} + 33$ | $12$ | $1$ | $11$ | $D_{12}$ | $[\ ]_{12}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e2.2t1.1c1 | $1$ | $ 2^{2}$ | $x^{2} + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e2_11.2t1.1c1 | $1$ | $ 2^{2} \cdot 11 $ | $x^{2} - 11$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.11.2t1.1c1 | $1$ | $ 11 $ | $x^{2} - x + 3$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.2e2_11e2.3t2.1c1 | $2$ | $ 2^{2} \cdot 11^{2}$ | $x^{3} - x^{2} + 4 x + 2$ | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2e2_11e2.6t3.1c1 | $2$ | $ 2^{2} \cdot 11^{2}$ | $x^{6} - x^{5} + 5 x^{4} - 6 x^{3} + 9 x^{2} - 5 x + 9$ | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.2e4_11e2.4t3.2c1 | $2$ | $ 2^{4} \cdot 11^{2}$ | $x^{4} + 11$ | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.2e4_11e2.12t12.2c1 | $2$ | $ 2^{4} \cdot 11^{2}$ | $x^{12} + 22 x^{8} - 55 x^{4} + 176$ | $D_{12}$ (as 12T12) | $1$ | $0$ |
| * | 2.2e4_11e2.12t12.2c2 | $2$ | $ 2^{4} \cdot 11^{2}$ | $x^{12} + 22 x^{8} - 55 x^{4} + 176$ | $D_{12}$ (as 12T12) | $1$ | $0$ |