Normalized defining polynomial
\( x^{10} - 4x^{9} + 2x^{8} + 12x^{7} + 5x^{6} - 30x^{5} - 103x^{4} - 142x^{3} - 112x^{2} - 48x - 10 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(221050053177344\) \(\medspace = 2^{12}\cdot 3779^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.19\) | 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: | $2^{7/4}3779^{1/2}\approx 206.7716238212183$ | ||
Ramified primes: | \(2\), \(3779\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{3779}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{14873}a^{9}-\frac{5132}{14873}a^{8}+\frac{6561}{14873}a^{7}-\frac{2070}{14873}a^{6}-\frac{4357}{14873}a^{5}+\frac{3420}{14873}a^{4}-\frac{2596}{14873}a^{3}+\frac{811}{14873}a^{2}+\frac{5520}{14873}a-\frac{3289}{14873}$
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: | $\frac{3542}{14873}a^{9}-\frac{17611}{14873}a^{8}+\frac{22309}{14873}a^{7}+\frac{30195}{14873}a^{6}-\frac{24066}{14873}a^{5}-\frac{97093}{14873}a^{4}-\frac{256359}{14873}a^{3}-\frac{206149}{14873}a^{2}-\frac{80520}{14873}a-\frac{4079}{14873}$, $\frac{115634}{14873}a^{9}-\frac{491797}{14873}a^{8}+\frac{330150}{14873}a^{7}+\frac{1431490}{14873}a^{6}+\frac{50156}{14873}a^{5}-\frac{3678421}{14873}a^{4}-\frac{10846522}{14873}a^{3}-\frac{12949474}{14873}a^{2}-\frac{7813186}{14873}a-\frac{2084963}{14873}$, $\frac{753}{14873}a^{9}+\frac{2584}{14873}a^{8}-\frac{27149}{14873}a^{7}+\frac{47574}{14873}a^{6}+\frac{50731}{14873}a^{5}-\frac{72134}{14873}a^{4}-\frac{244393}{14873}a^{3}-\frac{460173}{14873}a^{2}-\frac{305340}{14873}a-\frac{96937}{14873}$, $\frac{104556}{14873}a^{9}-\frac{498980}{14873}a^{8}+\frac{569711}{14873}a^{7}+\frac{937975}{14873}a^{6}-\frac{362327}{14873}a^{5}-\frac{3044111}{14873}a^{4}-\frac{8294260}{14873}a^{3}-\frac{7744890}{14873}a^{2}-\frac{3909244}{14873}a-\frac{541479}{14873}$, $\frac{6896}{14873}a^{9}-\frac{22278}{14873}a^{8}-\frac{13883}{14873}a^{7}+\frac{122344}{14873}a^{6}+\frac{71953}{14873}a^{5}-\frac{242226}{14873}a^{4}-\frac{887304}{14873}a^{3}-\frac{1308416}{14873}a^{2}-\frac{871594}{14873}a-\frac{297079}{14873}$ | 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: | \( 5909.00916583 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 5909.00916583 \cdot 1}{2\cdot\sqrt{221050053177344}}\cr\approx \mathstrut & 1.23885027302 \end{aligned}\]
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 fields
Degree 6 siblings: | 6.2.241856.1, 6.2.13815628323584.4 |
Degree 12 siblings: | data not computed |
Degree 15 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.241856.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
2.6.6.6 | $x^{6} - 4 x^{5} + 30 x^{4} - 16 x^{3} + 164 x^{2} + 160 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
\(3779\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |