Normalized defining polynomial
\( x^{6} - x^{5} - 6x^{4} + 2x^{2} - 12x - 3 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(23641797\) \(\medspace = 3\cdot 199^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.94\) | 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: | $3^{1/2}199^{1/2}\approx 24.43358344574123$ | ||
Ramified primes: | \(3\), \(199\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{597}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{29}a^{5}-\frac{11}{29}a^{4}-\frac{12}{29}a^{3}+\frac{4}{29}a^{2}-\frac{9}{29}a-\frac{9}{29}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | 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: | $\frac{3}{29}a^{5}-\frac{4}{29}a^{4}-\frac{7}{29}a^{3}-\frac{17}{29}a^{2}-\frac{27}{29}a+\frac{2}{29}$, $\frac{4}{29}a^{5}-\frac{15}{29}a^{4}+\frac{10}{29}a^{3}+\frac{16}{29}a^{2}-\frac{36}{29}a+\frac{22}{29}$, $\frac{3}{29}a^{5}-\frac{4}{29}a^{4}-\frac{7}{29}a^{3}+\frac{12}{29}a^{2}+\frac{2}{29}a+\frac{2}{29}$ | 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: | \( 26.4526843306 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 26.4526843306 \cdot 1}{2\cdot\sqrt{23641797}}\cr\approx \mathstrut & 0.429555456174 \end{aligned}\]
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
3.1.199.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q(\sqrt{-3}) \) $\times$ 4.2.1791.1 |
Degree 6 sibling: | 6.0.118803.1 |
Degree 8 siblings: | 8.0.3207681.1, 8.4.127027375281.1 |
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: | 6.0.118803.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(199\) | 199.2.1.2 | $x^{2} + 199$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
199.4.2.1 | $x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.199.2t1.a.a | $1$ | $ 199 $ | \(\Q(\sqrt{-199}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.597.2t1.a.a | $1$ | $ 3 \cdot 199 $ | \(\Q(\sqrt{597}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.1791.6t3.b.a | $2$ | $ 3^{2} \cdot 199 $ | 6.0.1069227.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.199.3t2.a.a | $2$ | $ 199 $ | 3.1.199.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.1791.4t5.b.a | $3$ | $ 3^{2} \cdot 199 $ | 4.2.1791.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
* | 3.118803.6t11.a.a | $3$ | $ 3 \cdot 199^{2}$ | 6.2.23641797.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ |
3.597.6t11.a.a | $3$ | $ 3 \cdot 199 $ | 6.2.23641797.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.356409.6t8.b.a | $3$ | $ 3^{2} \cdot 199^{2}$ | 4.2.1791.1 | $S_4$ (as 4T5) | $1$ | $-1$ |