Normalized defining polynomial
\( x^{6} - 3x^{5} + 84x^{4} - 161x^{3} + 2607x^{2} - 2706x + 29699 \)
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: | \(-9978460875\) \(\medspace = -\,3^{8}\cdot 5^{3}\cdot 23^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(46.40\) | 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: | $3^{4/3}5^{1/2}23^{1/2}\approx 46.39921073529938$ | ||
Ramified primes: | \(3\), \(5\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-115}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1035=3^{2}\cdot 5\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1035}(1,·)$, $\chi_{1035}(691,·)$, $\chi_{1035}(229,·)$, $\chi_{1035}(919,·)$, $\chi_{1035}(346,·)$, $\chi_{1035}(574,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-115}) \), 6.0.9978460875.4$^{3}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1768321}a^{5}-\frac{97410}{1768321}a^{4}-\frac{394532}{1768321}a^{3}-\frac{741930}{1768321}a^{2}-\frac{332832}{1768321}a-\frac{233296}{1768321}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{62}$, which has order $62$
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: | $\frac{18}{1768321}a^{5}+\frac{14941}{1768321}a^{4}-\frac{28292}{1768321}a^{3}+\frac{791828}{1768321}a^{2}-\frac{686013}{1768321}a+\frac{13483882}{1768321}$, $\frac{708}{1768321}a^{5}-\frac{1761}{1768321}a^{4}+\frac{66062}{1768321}a^{3}-\frac{95103}{1768321}a^{2}+\frac{1309958}{1768321}a+\frac{1048606}{1768321}$ | 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: | \( 3.39714980258 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 3.39714980258 \cdot 62}{2\cdot\sqrt{9978460875}}\cr\approx \mathstrut & 0.261507541421 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{-115}) \), \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{-115}) \) $\times$ \(\Q(\zeta_{9})^+\) |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | R | R | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.1.0.1}{1} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |
\(5\) | 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(23\) | 23.6.3.1 | $x^{6} + 1058 x^{2} - 219006$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.115.2t1.a.a | $1$ | $ 5 \cdot 23 $ | \(\Q(\sqrt{-115}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.1035.6t1.a.a | $1$ | $ 3^{2} \cdot 5 \cdot 23 $ | 6.0.9978460875.4 | $C_6$ (as 6T1) | $0$ | $-1$ |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.1035.6t1.a.b | $1$ | $ 3^{2} \cdot 5 \cdot 23 $ | 6.0.9978460875.4 | $C_6$ (as 6T1) | $0$ | $-1$ |