Normalized defining polynomial
\( x^{6} - x^{5} - 20x^{4} + 45x^{3} + 30x^{2} - 135x + 81 \)
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: | \(44851536\) \(\medspace = 2^{4}\cdot 3^{3}\cdot 47^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.85\) | 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: | $2^{2/3}3^{1/2}47^{1/2}\approx 18.849343120394256$ | ||
Ramified primes: | \(2\), \(3\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{141}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{3}a$
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: | $\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{28}{9}a^{3}-\frac{1}{3}a^{2}+\frac{14}{3}a-2$, $\frac{5}{9}a^{5}+\frac{4}{9}a^{4}-\frac{91}{9}a^{3}+7a^{2}+\frac{83}{3}a-26$, $\frac{4}{9}a^{5}-\frac{1}{9}a^{4}-\frac{83}{9}a^{3}+\frac{37}{3}a^{2}+\frac{79}{3}a-37$, $a-1$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{20}{9}a^{3}+4a^{2}+\frac{13}{3}a-8$ | 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: | \( 153.158171131 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 153.158171131 \cdot 1}{2\cdot\sqrt{44851536}}\cr\approx \mathstrut & 0.731815308057 \end{aligned}\]
Galois group
A solvable group of order 6 |
The 3 conjugacy class representatives for $S_3$ |
Character table for $S_3$ |
Intermediate fields
\(\Q(\sqrt{141}) \), 3.3.564.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.564.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
Degree 3 sibling: | 3.3.564.1 |
Minimal sibling: | 3.3.564.1 |
Multiplicative Galois module structure
$U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A'$ $\oplus$ $(A,\textrm{Sign})$ |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(47\) | 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |