Normalized defining polynomial
\( x^{6} - 2x^{5} - 39x^{4} - 32x^{3} + 126x^{2} + 56x - 88 \)
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: | \(2650991104\) \(\medspace = 2^{9}\cdot 173^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.20\) | 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: | $2^{3/2}173^{1/2}\approx 37.20215047547655$ | ||
Ramified primes: | \(2\), \(173\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{346}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2264}a^{5}-\frac{29}{283}a^{4}-\frac{1015}{2264}a^{3}+\frac{113}{1132}a^{2}+\frac{109}{1132}a-\frac{69}{566}$
Monogenic: | No | |
Index: | $8$ | |
Inessential primes: | $2$ |
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: | $\frac{643}{2264}a^{5}-\frac{221}{566}a^{4}-\frac{25517}{2264}a^{3}-\frac{18467}{1132}a^{2}+\frac{27071}{1132}a+\frac{17893}{566}$, $\frac{25}{2264}a^{5}-\frac{35}{566}a^{4}-\frac{471}{2264}a^{3}+\frac{1127}{1132}a^{2}+\frac{461}{1132}a-\frac{593}{566}$, $\frac{333}{2264}a^{5}-\frac{353}{566}a^{4}-\frac{9715}{2264}a^{3}+\frac{5367}{1132}a^{2}+\frac{6865}{1132}a-\frac{2601}{566}$, $\frac{241}{2264}a^{5}-\frac{111}{566}a^{4}-\frac{9159}{2264}a^{3}-\frac{5029}{1132}a^{2}+\frac{10421}{1132}a+\frac{3747}{566}$, $\frac{59}{566}a^{5}-\frac{52}{283}a^{4}-\frac{2153}{566}a^{3}-\frac{1540}{283}a^{2}+\frac{1054}{283}a+\frac{1763}{283}$ | 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: | \( 1689.96915243 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 1689.96915243 \cdot 2}{2\cdot\sqrt{2650991104}}\cr\approx \mathstrut & 2.10065443909 \end{aligned}\]
Galois group
A solvable group of order 6 |
The 3 conjugacy class representatives for $S_3$ |
Character table for $S_3$ |
Intermediate fields
\(\Q(\sqrt{346}) \), 3.3.1384.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.1384.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
Degree 3 sibling: | 3.3.1384.1 |
Minimal sibling: | 3.3.1384.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
\(173\) | 173.2.1.2 | $x^{2} + 346$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
173.2.1.2 | $x^{2} + 346$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
173.2.1.2 | $x^{2} + 346$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |