Normalized defining polynomial
\( x^{6} - x^{5} - 26x^{4} + 17x^{3} + 194x^{2} - 64x - 344 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(66961489\) \(\medspace = 7^{4}\cdot 167^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(20.15\) | 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: | $7^{2/3}167^{1/2}\approx 47.28865141512203$ | ||
Ramified primes: | \(7\), \(167\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{1204}a^{5}+\frac{163}{1204}a^{4}+\frac{109}{602}a^{3}-\frac{351}{1204}a^{2}+\frac{211}{602}a+\frac{3}{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{15}{1204}a^{5}+\frac{37}{1204}a^{4}-\frac{171}{602}a^{3}-\frac{449}{1204}a^{2}+\frac{757}{602}a+\frac{3}{7}$, $\frac{9}{602}a^{5}-\frac{19}{301}a^{4}-\frac{145}{602}a^{3}+\frac{453}{602}a^{2}+\frac{487}{602}a-\frac{2}{7}$, $\frac{9}{1204}a^{5}-\frac{339}{1204}a^{4}+\frac{39}{301}a^{3}+\frac{5269}{1204}a^{2}-\frac{405}{301}a-\frac{64}{7}$, $\frac{13}{602}a^{5}+\frac{6}{301}a^{4}-\frac{477}{602}a^{3}+\frac{253}{602}a^{2}+\frac{3981}{602}a-\frac{62}{7}$, $\frac{211}{1204}a^{5}+\frac{681}{1204}a^{4}-\frac{1683}{602}a^{3}-\frac{10249}{1204}a^{2}+\frac{4789}{602}a+\frac{136}{7}$ | 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: | \( 92.192814546 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 92.192814546 \cdot 1}{2\cdot\sqrt{66961489}}\cr\approx \mathstrut & 0.36052426561 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 4 conjugacy class representatives for $A_4$ |
Character table for $A_4$ |
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: | 12.12.125049841902709607569.1 |
Twin sextic algebra: | 4.4.1366561.1 $\times$ \(\Q\) $\times$ \(\Q\) |
Degree 4 sibling: | 4.4.1366561.1 |
Minimal sibling: | 4.4.1366561.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(167\) | 167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
167.2.0.1 | $x^{2} + 166 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
167.2.1.1 | $x^{2} + 835$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |