Normalized defining polynomial
\( x^{6} + x^{4} - 5x^{3} + 12x^{2} - 5x + 3 \)
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: | \(-3276875\) \(\medspace = -\,5^{4}\cdot 7^{2}\cdot 107\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.19\) | 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: | $5^{2/3}7^{1/2}107^{1/2}\approx 80.02412086555047$ | ||
Ramified primes: | \(5\), \(7\), \(107\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-107}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{4}+\frac{2}{9}a^{3}+\frac{2}{9}a^{2}+\frac{1}{9}a+\frac{1}{3}$
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: | $\frac{1}{3}a^{5}-2a^{2}+\frac{11}{3}a$, $\frac{1}{9}a^{5}+\frac{5}{9}a^{4}+\frac{8}{9}a^{3}+\frac{8}{9}a^{2}-\frac{2}{9}a+\frac{1}{3}$ | 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: | \( 11.2231204428 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 11.2231204428 \cdot 1}{2\cdot\sqrt{3276875}}\cr\approx \mathstrut & 0.768941491130 \end{aligned}\]
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
3.1.175.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q(\sqrt{749}) \) $\times$ 4.2.2003575.2 |
Degree 6 sibling: | 6.2.22938125.1 |
Degree 8 siblings: | 8.4.196701326250625.3, 8.0.4014312780625.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(107\) | $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.749.2t1.a.a | $1$ | $ 7 \cdot 107 $ | \(\Q(\sqrt{749}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.107.2t1.a.a | $1$ | $ 107 $ | \(\Q(\sqrt{-107}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.2003575.6t3.a.a | $2$ | $ 5^{2} \cdot 7 \cdot 107^{2}$ | 6.2.262618593125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.175.3t2.b.a | $2$ | $ 5^{2} \cdot 7 $ | 3.1.175.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.2003575.4t5.b.a | $3$ | $ 5^{2} \cdot 7 \cdot 107^{2}$ | 4.2.2003575.2 | $S_4$ (as 4T5) | $1$ | $1$ | |
* | 3.18725.6t11.f.a | $3$ | $ 5^{2} \cdot 7 \cdot 107 $ | 6.0.3276875.3 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
3.131075.6t11.f.a | $3$ | $ 5^{2} \cdot 7^{2} \cdot 107 $ | 6.0.3276875.3 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.14025025.6t8.c.a | $3$ | $ 5^{2} \cdot 7^{2} \cdot 107^{2}$ | 4.2.2003575.2 | $S_4$ (as 4T5) | $1$ | $-1$ |