Normalized defining polynomial
\( x^{6} - 12x^{4} - 5x^{3} + 36x^{2} + 30x + 1 \)
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: | \(6751269\) \(\medspace = 3^{9}\cdot 7^{3}\) | 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: | \(13.75\) | 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: | $3^{3/2}7^{1/2}\approx 13.74772708486752$ | ||
Ramified primes: | \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(63=3^{2}\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{63}(1,·)$, $\chi_{63}(20,·)$, $\chi_{63}(22,·)$, $\chi_{63}(41,·)$, $\chi_{63}(43,·)$, $\chi_{63}(62,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{17}a^{5}+\frac{6}{17}a^{4}+\frac{7}{17}a^{3}+\frac{3}{17}a^{2}+\frac{3}{17}a-\frac{3}{17}$
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: | $\frac{3}{17}a^{5}+\frac{1}{17}a^{4}-\frac{30}{17}a^{3}-\frac{8}{17}a^{2}+\frac{60}{17}a+\frac{8}{17}$, $\frac{2}{17}a^{5}-\frac{5}{17}a^{4}-\frac{20}{17}a^{3}+\frac{23}{17}a^{2}+\frac{57}{17}a+\frac{11}{17}$, $\frac{4}{17}a^{5}-\frac{10}{17}a^{4}-\frac{40}{17}a^{3}+\frac{63}{17}a^{2}+\frac{97}{17}a+\frac{5}{17}$, $\frac{2}{17}a^{5}-\frac{5}{17}a^{4}-\frac{20}{17}a^{3}+\frac{23}{17}a^{2}+\frac{74}{17}a+\frac{11}{17}$, $\frac{1}{17}a^{5}+\frac{6}{17}a^{4}-\frac{10}{17}a^{3}-\frac{48}{17}a^{2}+\frac{3}{17}a+\frac{31}{17}$ | 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: | \( 26.8507980102 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 26.8507980102 \cdot 1}{2\cdot\sqrt{6751269}}\cr\approx \mathstrut & 0.330684980201 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{21}) \) $\times$ \(\Q(\zeta_{9})^+\) |
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 | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.1.0.1}{1} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
\(7\) | 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.63.6t1.h.a | $1$ | $ 3^{2} \cdot 7 $ | 6.6.6751269.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.63.6t1.h.b | $1$ | $ 3^{2} \cdot 7 $ | 6.6.6751269.1 | $C_6$ (as 6T1) | $0$ | $1$ |