Normalized defining polynomial
\( x^{6} - 106x^{4} - 404x^{3} + 228x^{2} + 764x - 492 \)
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: | \(275095823056\) \(\medspace = 2^{4}\cdot 29^{3}\cdot 89^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(80.65\) | 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: | $2^{2/3}29^{1/2}89^{1/2}\approx 80.6455978932349$ | ||
Ramified primes: | \(2\), \(29\), \(89\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2581}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{6}a^{3}+\frac{1}{3}a$, $\frac{1}{18}a^{4}+\frac{1}{18}a^{3}-\frac{2}{9}a^{2}+\frac{4}{9}a-\frac{1}{3}$, $\frac{1}{324}a^{5}+\frac{2}{81}a^{4}+\frac{1}{27}a^{3}+\frac{35}{162}a^{2}-\frac{19}{81}a-\frac{5}{27}$
Monogenic: | No | |
Index: | $27$ | |
Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{7}{81}a^{5}-\frac{23}{162}a^{4}-\frac{242}{27}a^{3}-\frac{1670}{81}a^{2}+\frac{4166}{81}a-\frac{599}{27}$, $\frac{35}{81}a^{5}+\frac{11}{162}a^{4}-\frac{2351}{54}a^{3}-\frac{15649}{81}a^{2}-\frac{7745}{81}a+\frac{5114}{27}$, $\frac{31}{324}a^{5}-\frac{47}{162}a^{4}-\frac{481}{54}a^{3}-\frac{2281}{162}a^{2}+\frac{3344}{81}a-\frac{497}{27}$, $\frac{1}{4}a^{5}-\frac{1}{18}a^{4}-\frac{451}{18}a^{3}-\frac{1859}{18}a^{2}-\frac{202}{9}a+\frac{430}{3}$, $\frac{1}{18}a^{4}+\frac{7}{18}a^{3}-\frac{2}{9}a^{2}-\frac{8}{9}a+\frac{2}{3}$ | 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: | \( 21603.5776536 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 21603.5776536 \cdot 2}{2\cdot\sqrt{275095823056}}\cr\approx \mathstrut & 2.63611076201 \end{aligned}\]
Galois group
A solvable group of order 6 |
The 3 conjugacy class representatives for $S_3$ |
Character table for $S_3$ |
Intermediate fields
\(\Q(\sqrt{2581}) \), 3.3.10324.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.10324.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
Degree 3 sibling: | 3.3.10324.1 |
Minimal sibling: | 3.3.10324.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.1.0.1}{1} }^{6}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
\(29\) | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(89\) | 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |