Normalized defining polynomial
\( x^{10} - 5x^{8} - 2x^{7} + 9x^{6} - 11x^{4} + 17x^{3} + 2x^{2} - 10x - 3 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-21883862619611\) \(\medspace = -\,83^{3}\cdot 337^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.58\) | 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: | $83^{1/2}337^{1/2}\approx 167.24532878379594$ | ||
Ramified primes: | \(83\), \(337\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-27971}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{23}a^{9}-\frac{10}{23}a^{8}+\frac{3}{23}a^{7}-\frac{9}{23}a^{6}+\frac{7}{23}a^{5}-\frac{1}{23}a^{4}-\frac{1}{23}a^{3}+\frac{4}{23}a^{2}+\frac{8}{23}a+\frac{2}{23}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{394}{23}a^{9}-\frac{168}{23}a^{8}-\frac{1900}{23}a^{7}+\frac{19}{23}a^{6}+\frac{3540}{23}a^{5}-\frac{1498}{23}a^{4}-\frac{3683}{23}a^{3}+\frac{8269}{23}a^{2}-\frac{2713}{23}a-\frac{2777}{23}$, $\frac{497}{23}a^{9}-\frac{209}{23}a^{8}-\frac{2396}{23}a^{7}+\frac{12}{23}a^{6}+\frac{4468}{23}a^{5}-\frac{1877}{23}a^{4}-\frac{4683}{23}a^{3}+\frac{10406}{23}a^{2}-\frac{3361}{23}a-\frac{3560}{23}$, $\frac{206}{23}a^{9}-\frac{82}{23}a^{8}-\frac{992}{23}a^{7}-\frac{14}{23}a^{6}+\frac{1833}{23}a^{5}-\frac{758}{23}a^{4}-\frac{1931}{23}a^{3}+\frac{4297}{23}a^{2}-\frac{1319}{23}a-\frac{1451}{23}$, $\frac{1324}{23}a^{9}-\frac{567}{23}a^{8}-\frac{6378}{23}a^{7}+\frac{90}{23}a^{6}+\frac{11890}{23}a^{5}-\frac{5119}{23}a^{4}-\frac{12433}{23}a^{3}+\frac{27859}{23}a^{2}-\frac{9188}{23}a-\frac{9335}{23}$, $\frac{38}{23}a^{9}-\frac{12}{23}a^{8}-\frac{185}{23}a^{7}-\frac{20}{23}a^{6}+\frac{335}{23}a^{5}-\frac{107}{23}a^{4}-\frac{360}{23}a^{3}+\frac{773}{23}a^{2}-\frac{156}{23}a-\frac{223}{23}$, $\frac{82}{23}a^{9}-\frac{38}{23}a^{8}-\frac{398}{23}a^{7}+\frac{21}{23}a^{6}+\frac{758}{23}a^{5}-\frac{335}{23}a^{4}-\frac{818}{23}a^{3}+\frac{1731}{23}a^{2}-\frac{540}{23}a-\frac{572}{23}$ | 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: | \( 2446.33330103 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 2446.33330103 \cdot 1}{2\cdot\sqrt{21883862619611}}\cr\approx \mathstrut & 1.03772726979 \end{aligned}\]
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: | 6.0.27971.1, 6.4.21883862619611.1 |
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: | 6.0.27971.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(83\) | $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
83.3.0.1 | $x^{3} + 3 x + 81$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
83.6.3.1 | $x^{6} + 20667 x^{2} - 46314747$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(337\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |