Normalized defining polynomial
\( x^{11} - 2x^{10} + 3x^{9} - 3x^{8} + 3x^{7} - 3x^{6} + 5x^{5} - 9x^{4} + 11x^{3} - 8x^{2} + 4x - 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-12696946299\) \(\medspace = -\,3^{2}\cdot 15731\cdot 89681\) | 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: | \(8.29\) | 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: | $3^{1/2}15731^{1/2}89681^{1/2}\approx 65056.248224132934$ | ||
Ramified primes: | \(3\), \(15731\), \(89681\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-1410771811}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23}a^{10}-\frac{11}{23}a^{9}+\frac{10}{23}a^{8}-\frac{1}{23}a^{7}-\frac{11}{23}a^{6}+\frac{4}{23}a^{5}-\frac{8}{23}a^{4}-\frac{6}{23}a^{3}-\frac{4}{23}a^{2}+\frac{5}{23}a+\frac{5}{23}$
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)
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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)
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Fundamental units: | $\frac{41}{23}a^{10}-\frac{60}{23}a^{9}+\frac{88}{23}a^{8}-\frac{64}{23}a^{7}+\frac{78}{23}a^{6}-\frac{66}{23}a^{5}+\frac{155}{23}a^{4}-\frac{269}{23}a^{3}+\frac{273}{23}a^{2}-\frac{140}{23}a+\frac{44}{23}$, $a$, $\frac{42}{23}a^{10}-\frac{71}{23}a^{9}+\frac{98}{23}a^{8}-\frac{88}{23}a^{7}+\frac{90}{23}a^{6}-\frac{85}{23}a^{5}+\frac{170}{23}a^{4}-\frac{321}{23}a^{3}+\frac{338}{23}a^{2}-\frac{204}{23}a+\frac{72}{23}$, $\frac{51}{23}a^{10}-\frac{78}{23}a^{9}+\frac{119}{23}a^{8}-\frac{97}{23}a^{7}+\frac{106}{23}a^{6}-\frac{95}{23}a^{5}+\frac{213}{23}a^{4}-\frac{352}{23}a^{3}+\frac{394}{23}a^{2}-\frac{228}{23}a+\frac{94}{23}$, $\frac{11}{23}a^{10}-\frac{29}{23}a^{9}+\frac{41}{23}a^{8}-\frac{34}{23}a^{7}+\frac{40}{23}a^{6}-\frac{25}{23}a^{5}+\frac{73}{23}a^{4}-\frac{112}{23}a^{3}+\frac{163}{23}a^{2}-\frac{83}{23}a+\frac{32}{23}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 2.30111811917 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 2.30111811917 \cdot 1}{2\cdot\sqrt{12696946299}}\cr\approx \mathstrut & 0.199980943260 \end{aligned}\]
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | R | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.7.0.1 | $x^{7} + 2 x^{2} + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(15731\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(89681\) | $\Q_{89681}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{89681}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{89681}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{89681}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |