Normalized defining polynomial
\( x^{6} - x^{5} + 3 x^{4} - 9 x^{3} + 7 x^{2} - 4 x - 2 \)
Invariants
Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
Signature: | $[2, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
Discriminant: | \(2287148=2^{2}\cdot 83^{3}\) | magma: Discriminant(K);
sage: K.disc()
gp: K.disc
| |
Root discriminant: | $11.48$ | magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
Ramified primes: | $2, 83$ | magma: PrimeDivisors(Discriminant(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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{23} a^{5} + \frac{6}{23} a^{4} - \frac{1}{23} a^{3} + \frac{7}{23} a^{2} + \frac{10}{23} a - \frac{3}{23}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | 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: | \( \frac{6}{23} a^{5} - \frac{10}{23} a^{4} + \frac{17}{23} a^{3} - \frac{73}{23} a^{2} + \frac{60}{23} a - \frac{41}{23} \), \( \frac{14}{23} a^{5} - \frac{8}{23} a^{4} + \frac{32}{23} a^{3} - \frac{109}{23} a^{2} + \frac{48}{23} a + \frac{27}{23} \), \( \frac{10}{23} a^{5} - \frac{9}{23} a^{4} + \frac{36}{23} a^{3} - \frac{91}{23} a^{2} + \frac{77}{23} a - \frac{99}{23} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
Regulator: | \( 13.0560185162 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times S_4$ (as 6T11):
A solvable group of order 48 |
The 10 conjugacy class representatives for $S_4\times C_2$ |
Character table for $S_4\times C_2$ |
Intermediate fields
3.1.83.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 4.2.1328.1 $\times$ \(\Q(\sqrt{-1}) \) |
Degree 6 sibling: | 6.0.27556.1 |
Degree 8 siblings: | 8.4.12149330176.1, 8.0.1763584.1 |
Degree 12 siblings: | 12.2.1008394404608.1, 12.0.194389282816.1, 12.0.5231045973904.1, 12.4.1339147769319424.1, 12.0.1339147769319424.1, 12.0.1339147769319424.3 |
Degree 16 sibling: | Deg 16 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }$ | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
83 | Data not computed |
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) | $1$ | $-1$ | |
1.83.2t1.1c1 | $1$ | $ 83 $ | $x^{2} - x + 21$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2e2_83.2t1.1c1 | $1$ | $ 2^{2} \cdot 83 $ | $x^{2} - 83$ | $C_2$ (as 2T1) | $1$ | $1$ | |
2.2e4_83.6t3.1c1 | $2$ | $ 2^{4} \cdot 83 $ | $x^{6} - x^{4} - 3 x^{2} + 4$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.83.3t2.1c1 | $2$ | $ 83 $ | $x^{3} - x^{2} + x - 2$ | $S_3$ (as 3T2) | $1$ | $0$ |
3.2e4_83.4t5.1c1 | $3$ | $ 2^{4} \cdot 83 $ | $x^{4} - 3 x^{2} - 2 x + 1$ | $S_4$ (as 4T5) | $1$ | $1$ | |
* | 3.2e2_83e2.6t11.1c1 | $3$ | $ 2^{2} \cdot 83^{2}$ | $x^{6} - x^{5} + 3 x^{4} - 9 x^{3} + 7 x^{2} - 4 x - 2$ | $S_4\times C_2$ (as 6T11) | $1$ | $1$ |
3.2e2_83.6t11.1c1 | $3$ | $ 2^{2} \cdot 83 $ | $x^{6} - x^{5} + 3 x^{4} - 9 x^{3} + 7 x^{2} - 4 x - 2$ | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.2e4_83e2.6t8.2c1 | $3$ | $ 2^{4} \cdot 83^{2}$ | $x^{4} - 3 x^{2} - 2 x + 1$ | $S_4$ (as 4T5) | $1$ | $-1$ |