Normalized defining polynomial
\( x^{6} - 2x^{3} - 3x^{2} + 4x - 1 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-217024\) \(\medspace = -\,2^{6}\cdot 3391\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.75\) | 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: | $2^{3/2}3391^{1/2}\approx 164.7057983193063$ | ||
Ramified primes: | \(2\), \(3391\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3391}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $a$, $4a^{5}+2a^{4}+a^{3}-8a^{2}-16a+8$, $4a^{5}+2a^{4}+a^{3}-7a^{2}-16a+8$, $2a^{5}+a^{4}-4a^{2}-9a+5$ | 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: | \( 2.36664754095 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 2.36664754095 \cdot 1}{2\cdot\sqrt{217024}}\cr\approx \mathstrut & 0.255358052664 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | 6.0.2495533150144.1 |
Degree 6 sibling: | 6.0.2495533150144.1 |
Degree 10 sibling: | 10.4.159714121609216.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 siblings: | 20.8.25508600641403437249416134656.1, deg 20, deg 20 |
Degree 30 siblings: | deg 30, deg 30, deg 30, deg 30, deg 30, deg 30 |
Degree 36 sibling: | deg 36 |
Degree 40 siblings: | deg 40, deg 40, deg 40 |
Degree 45 sibling: | deg 45 |
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 | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
\(3391\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3391.2t1.a.a | $1$ | $ 3391 $ | \(\Q(\sqrt{-3391}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 5.217024.6t16.a.a | $5$ | $ 2^{6} \cdot 3391 $ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $3$ |
5.735928384.12t183.a.a | $5$ | $ 2^{6} \cdot 3391^{2}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
5.846...304.12t183.a.a | $5$ | $ 2^{6} \cdot 3391^{4}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
5.249...144.6t16.a.a | $5$ | $ 2^{6} \cdot 3391^{3}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $-1$ | |
9.159...216.10t32.a.a | $9$ | $ 2^{12} \cdot 3391^{3}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
9.622...736.20t145.a.a | $9$ | $ 2^{12} \cdot 3391^{6}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
10.398...104.30t164.a.a | $10$ | $ 2^{18} \cdot 3391^{6}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
10.346...184.30t164.a.a | $10$ | $ 2^{18} \cdot 3391^{4}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $2$ | |
16.293...936.36t1252.a.a | $16$ | $ 2^{24} \cdot 3391^{8}$ | 6.4.217024.1 | $S_6$ (as 6T16) | $1$ | $0$ |