Normalized defining polynomial
\( x^{6} - 2x^{5} - 130x^{4} - 840x^{3} + 7756x^{2} + 48632x + 88520 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(254390584509184\) \(\medspace = 2^{8}\cdot 17^{3}\cdot 587^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(251.72\) | 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: | $2^{7/4}17^{1/2}587^{1/2}\approx 336.00520399447527$ | ||
Ramified primes: | \(2\), \(17\), \(587\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{9979}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{36}a^{3}-\frac{1}{18}a^{2}-\frac{5}{18}a-\frac{4}{9}$, $\frac{1}{216}a^{4}+\frac{1}{108}a^{3}+\frac{13}{54}a-\frac{7}{54}$, $\frac{1}{648}a^{5}-\frac{1}{162}a^{3}+\frac{13}{162}a^{2}+\frac{7}{54}a+\frac{34}{81}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)
Unit group
Rank: | $3$ | 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: | $\frac{35363}{324}a^{5}+\frac{95591}{108}a^{4}-\frac{405269}{81}a^{3}-\frac{11661427}{81}a^{2}-\frac{16706603}{27}a-\frac{77621216}{81}$, $\frac{59}{648}a^{5}-\frac{151}{108}a^{4}+\frac{176}{81}a^{3}-\frac{1825}{162}a^{2}+\frac{6815}{18}a+\frac{6473}{81}$, $\frac{45425}{324}a^{5}-\frac{45877}{36}a^{4}-\frac{1005446}{81}a^{3}-\frac{1130429}{162}a^{2}+\frac{39607811}{27}a-\frac{223511921}{81}$ (assuming GRH) | 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: | \( 23711.9463916 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 23711.9463916 \cdot 16}{2\cdot\sqrt{254390584509184}}\cr\approx \mathstrut & 1.87813325643 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_6$ |
Character table for $S_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | 6.2.638656.1 |
Degree 6 sibling: | 6.2.638656.1 |
Degree 10 sibling: | deg 10 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 siblings: | deg 15, deg 15 |
Degree 20 siblings: | deg 20, deg 20, deg 20 |
Degree 30 siblings: | deg 30, deg 30, deg 30, deg 30, deg 30, deg 30 |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | deg 40, deg 40, some data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.638656.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
\(17\) | 17.6.3.1 | $x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(587\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $2$ | $2$ | $2$ |