Normalized defining polynomial
\( x^{11} - x^{10} - 6x^{9} + 5x^{8} + 10x^{7} - 8x^{6} + 5x^{4} - 10x^{3} - 2x^{2} + 4x + 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-7530807227563\) \(\medspace = -\,29\cdot 131\cdot 5399\cdot 367163\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(14.81\) | 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))
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Galois root discriminant: | $29^{1/2}131^{1/2}5399^{1/2}367163^{1/2}\approx 2744231.6278993287$ | ||
Ramified primes: | \(29\), \(131\), \(5399\), \(367163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-7530807227563}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $a$, $a^{2}-1$, $a^{9}-a^{8}-5a^{7}+4a^{6}+5a^{5}-4a^{4}+5a^{3}+a^{2}-6a-1$, $2a^{9}-4a^{8}-6a^{7}+14a^{6}-3a^{5}-6a^{4}+13a^{3}-9a^{2}-2a+1$, $2a^{10}-4a^{9}-8a^{8}+17a^{7}+4a^{6}-16a^{5}+13a^{4}-5a^{3}-13a^{2}+5a+3$, $a^{10}-a^{9}-5a^{8}+4a^{7}+5a^{6}-4a^{5}+5a^{4}+a^{3}-6a^{2}$, $a^{9}-2a^{8}-4a^{7}+8a^{6}+2a^{5}-6a^{4}+7a^{3}-3a^{2}-6a+1$, $a^{10}-2a^{9}-4a^{8}+9a^{7}+2a^{6}-10a^{5}+6a^{4}-2a^{3}-7a^{2}+5a+3$, $a^{10}-a^{9}-6a^{8}+5a^{7}+10a^{6}-8a^{5}+5a^{3}-10a^{2}-a+3$ | 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: | \( 302.713730235 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{1}\cdot 302.713730235 \cdot 1}{2\cdot\sqrt{7530807227563}}\cr\approx \mathstrut & 0.177431689565 \end{aligned}\]
Galois group
A non-solvable group of order 39916800 |
The 56 conjugacy class representatives for $S_{11}$ |
Character table for $S_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 22 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
29.5.0.1 | $x^{5} + 3 x + 27$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(131\) | 131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
131.5.0.1 | $x^{5} + 19 x + 129$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(5399\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(367163\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |