Normalized defining polynomial
\( x^{6} - 2x^{4} - 10x^{2} + 22 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2725888\) \(\medspace = -\,2^{11}\cdot 11^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.82\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{31/12}11^{1/2}\approx 19.877289578936544$ | ||
Ramified primes: | \(2\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-22}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $2$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a^{4}+a^{2}-9$, $3a^{5}+5a^{4}+2a^{3}+3a^{2}-25a-41$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 4.71451936461 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 4.71451936461 \cdot 2}{2\cdot\sqrt{2725888}}\cr\approx \mathstrut & 0.708309432173 \end{aligned}\]
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
3.1.44.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 4.2.2816.1 $\times$ \(\Q(\sqrt{2}) \) |
Degree 6 sibling: | 6.2.247808.2 |
Degree 8 siblings: | 8.4.126877696.1, 8.0.15352201216.3 |
Degree 12 siblings: | 12.2.43231798624256.2, 12.4.3930163511296.1, 12.0.475549784866816.15, 12.0.475549784866816.18, 12.0.7430465388544.5, 12.0.475549784866816.23 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
Minimal sibling: | 6.2.247808.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.11.8 | $x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $[8/3, 8/3, 3]_{3}^{2}$ |
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.88.2t1.b.a | $1$ | $ 2^{3} \cdot 11 $ | \(\Q(\sqrt{-22}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.704.6t3.d.a | $2$ | $ 2^{6} \cdot 11 $ | 6.2.247808.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.44.3t2.b.a | $2$ | $ 2^{2} \cdot 11 $ | 3.1.44.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.2816.4t5.c.a | $3$ | $ 2^{8} \cdot 11 $ | 4.2.2816.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.5632.6t11.c.a | $3$ | $ 2^{9} \cdot 11 $ | 6.0.2725888.2 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
* | 3.61952.6t11.m.a | $3$ | $ 2^{9} \cdot 11^{2}$ | 6.0.2725888.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
3.30976.6t8.g.a | $3$ | $ 2^{8} \cdot 11^{2}$ | 4.2.2816.1 | $S_4$ (as 4T5) | $1$ | $-1$ |