Normalized defining polynomial
\( x^{6} + 2x^{4} + 2x^{2} - 2x + 6 \)
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: | \(-716224\) \(\medspace = -\,2^{6}\cdot 19^{2}\cdot 31\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.46\) | 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: | $2^{7/6}19^{1/2}31^{1/2}\approx 54.482786213191424$ | ||
Ramified primes: | \(2\), \(19\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{19}a^{5}-\frac{9}{19}a^{4}+\frac{7}{19}a^{3}-\frac{6}{19}a^{2}-\frac{1}{19}a+\frac{7}{19}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{2}{19}a^{5}+\frac{1}{19}a^{4}-\frac{5}{19}a^{3}+\frac{7}{19}a^{2}-\frac{2}{19}a-\frac{5}{19}$, $\frac{8}{19}a^{5}+\frac{4}{19}a^{4}-\frac{1}{19}a^{3}-\frac{10}{19}a^{2}+\frac{11}{19}a-\frac{1}{19}$ | 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: | \( 6.81858341482 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 6.81858341482 \cdot 1}{2\cdot\sqrt{716224}}\cr\approx \mathstrut & 0.999261907947 \end{aligned}\]
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
3.1.76.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q(\sqrt{589}) \) $\times$ 4.2.292144.1 |
Degree 6 sibling: | 6.2.13608256.1 |
Degree 8 siblings: | 8.4.30810670141696.9, 8.0.85348116736.1 |
Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | R | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.7 | $x^{6} + 2 x^{2} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(31\) | 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.589.2t1.a.a | $1$ | $ 19 \cdot 31 $ | \(\Q(\sqrt{589}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.73036.6t3.b.a | $2$ | $ 2^{2} \cdot 19 \cdot 31^{2}$ | 6.2.3269383504.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.76.3t2.a.a | $2$ | $ 2^{2} \cdot 19 $ | 3.1.76.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.292144.4t5.a.a | $3$ | $ 2^{4} \cdot 19 \cdot 31^{2}$ | 4.2.292144.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
* | 3.9424.6t11.a.a | $3$ | $ 2^{4} \cdot 19 \cdot 31 $ | 6.0.716224.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
3.179056.6t11.a.a | $3$ | $ 2^{4} \cdot 19^{2} \cdot 31 $ | 6.0.716224.2 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.5550736.6t8.a.a | $3$ | $ 2^{4} \cdot 19^{2} \cdot 31^{2}$ | 4.2.292144.1 | $S_4$ (as 4T5) | $1$ | $-1$ |