Normalized defining polynomial
\( x^{6} - x^{5} + x^{4} - 8x^{3} - 6x^{2} - 15x - 13 \)
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)
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Discriminant: | \(2336173\) \(\medspace = 7^{5}\cdot 139\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.52\) | 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))
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Galois root discriminant: | $7^{5/6}139^{1/2}\approx 59.66996276060805$ | ||
Ramified primes: | \(7\), \(139\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{973}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{113}a^{5}+\frac{31}{113}a^{4}-\frac{24}{113}a^{3}+\frac{15}{113}a^{2}+\frac{22}{113}a+\frac{11}{113}$
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)
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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: | $\frac{3}{113}a^{5}-\frac{20}{113}a^{4}+\frac{41}{113}a^{3}-\frac{68}{113}a^{2}+\frac{66}{113}a+\frac{33}{113}$, $\frac{3}{113}a^{5}-\frac{20}{113}a^{4}+\frac{41}{113}a^{3}-\frac{68}{113}a^{2}+\frac{66}{113}a-\frac{80}{113}$, $\frac{4}{113}a^{5}+\frac{11}{113}a^{4}+\frac{17}{113}a^{3}+\frac{60}{113}a^{2}+\frac{88}{113}a+\frac{44}{113}$ | 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: | \( 5.9718100677 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 5.9718100677 \cdot 1}{2\cdot\sqrt{2336173}}\cr\approx \mathstrut & 0.30849151180 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 6T6):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4\times C_2$ |
Character table for $A_4\times C_2$ |
Intermediate fields
\(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Galois closure: | deg 24 |
Twin sextic algebra: | \(\Q(\sqrt{973}) \) $\times$ 4.0.946729.1 |
Degree 8 sibling: | 8.0.43918494172609.2 |
Degree 12 siblings: | deg 12, deg 12 |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
\(139\) | 139.2.0.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
139.2.0.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |