Normalized defining polynomial
\( x^{6} - x^{5} - 89x^{4} + 59x^{3} + 2385x^{2} - 841x - 18901 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(3464395697\) \(\medspace = 7^{4}\cdot 113^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.90\) | 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: | $7^{2/3}113^{1/2}\approx 38.898953270916685$ | ||
Ramified primes: | \(7\), \(113\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{113}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(791=7\cdot 113\) | ||
Dirichlet character group: | $\lbrace$$\chi_{791}(1,·)$, $\chi_{791}(338,·)$, $\chi_{791}(564,·)$, $\chi_{791}(225,·)$, $\chi_{791}(340,·)$, $\chi_{791}(114,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1269619}a^{5}+\frac{269602}{1269619}a^{4}-\frac{179833}{1269619}a^{3}-\frac{575487}{1269619}a^{2}-\frac{499000}{1269619}a-\frac{529363}{1269619}$
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)
| |
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: | $\frac{226}{1269619}a^{5}-\frac{11660}{1269619}a^{4}-\frac{14450}{1269619}a^{3}+\frac{710695}{1269619}a^{2}+\frac{222091}{1269619}a-\frac{9179185}{1269619}$, $\frac{438}{1269619}a^{5}+\frac{11109}{1269619}a^{4}-\frac{50476}{1269619}a^{3}-\frac{678744}{1269619}a^{2}+\frac{1082087}{1269619}a+\frac{8096997}{1269619}$, $\frac{54052}{1269619}a^{5}-\frac{159578}{1269619}a^{4}-\frac{3939109}{1269619}a^{3}+\frac{10868747}{1269619}a^{2}+\frac{67127843}{1269619}a-\frac{181280990}{1269619}$, $\frac{57861}{1269619}a^{5}-\frac{367331}{1269619}a^{4}-\frac{3328746}{1269619}a^{3}+\frac{20358110}{1269619}a^{2}+\frac{32776773}{1269619}a-\frac{200513970}{1269619}$, $\frac{96944}{1269619}a^{5}-\frac{80446}{1269619}a^{4}-\frac{9479196}{1269619}a^{3}+\frac{4664846}{1269619}a^{2}+\frac{375774362}{1269619}a-\frac{1050541605}{1269619}$ | 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: | \( 825.407118853 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 825.407118853 \cdot 1}{2\cdot\sqrt{3464395697}}\cr\approx \mathstrut & 0.448749981068 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{113}) \), \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{113}) \) $\times$ \(\Q\) |
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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(113\) | 113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.113.2t1.a.a | $1$ | $ 113 $ | \(\Q(\sqrt{113}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.791.6t1.a.a | $1$ | $ 7 \cdot 113 $ | 6.6.3464395697.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.791.6t1.a.b | $1$ | $ 7 \cdot 113 $ | 6.6.3464395697.1 | $C_6$ (as 6T1) | $0$ | $1$ |