Normalized defining polynomial
\( x^{11} - x^{10} - 4x^{9} + 7x^{8} + 4x^{7} - 9x^{6} - 5x^{5} + 2x^{4} + 8x^{3} + 2x^{2} - 3x - 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-145065457639\) \(\medspace = -\,2389\cdot 2617\cdot 23203\) | 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: | \(10.34\) | 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: | $2389^{1/2}2617^{1/2}23203^{1/2}\approx 380874.5956860342$ | ||
Ramified primes: | \(2389\), \(2617\), \(23203\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-145065457639}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{349}a^{10}+\frac{74}{349}a^{9}-\frac{38}{349}a^{8}-\frac{51}{349}a^{7}+\frac{18}{349}a^{6}-\frac{55}{349}a^{5}+\frac{58}{349}a^{4}+\frac{164}{349}a^{3}+\frac{93}{349}a^{2}-\frac{3}{349}a+\frac{121}{349}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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)
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Fundamental units: | $\frac{314}{349}a^{10}-\frac{496}{349}a^{9}-\frac{1113}{349}a^{8}+\frac{2832}{349}a^{7}+\frac{68}{349}a^{6}-\frac{3310}{349}a^{5}-\frac{285}{349}a^{4}+\frac{891}{349}a^{3}+\frac{2329}{349}a^{2}-\frac{244}{349}a-\frac{745}{349}$, $\frac{344}{349}a^{10}-\frac{370}{349}a^{9}-\frac{1206}{349}a^{8}+\frac{2349}{349}a^{7}+\frac{608}{349}a^{6}-\frac{2168}{349}a^{5}-\frac{988}{349}a^{4}-\frac{471}{349}a^{3}+\frac{1978}{349}a^{2}+\frac{713}{349}a-\frac{256}{349}$, $a^{10}-a^{9}-4a^{8}+7a^{7}+4a^{6}-9a^{5}-5a^{4}+2a^{3}+8a^{2}+2a-3$, $\frac{136}{349}a^{10}-\frac{57}{349}a^{9}-\frac{631}{349}a^{8}+\frac{742}{349}a^{7}+\frac{1052}{349}a^{6}-\frac{1198}{349}a^{5}-\frac{837}{349}a^{4}-\frac{32}{349}a^{3}+\frac{1131}{349}a^{2}+\frac{639}{349}a-\frac{296}{349}$, $\frac{427}{349}a^{10}-\frac{859}{349}a^{9}-\frac{1219}{349}a^{8}+\frac{4398}{349}a^{7}-\frac{1388}{349}a^{6}-\frac{4290}{349}a^{5}+\frac{685}{349}a^{4}+\frac{1624}{349}a^{3}+\frac{3066}{349}a^{2}-\frac{1281}{349}a-\frac{1032}{349}$, $\frac{583}{349}a^{10}-\frac{832}{349}a^{9}-\frac{1912}{349}a^{8}+\frac{4818}{349}a^{7}+\frac{24}{349}a^{6}-\frac{4843}{349}a^{5}-\frac{737}{349}a^{4}+\frac{1033}{349}a^{3}+\frac{3963}{349}a^{2}-\frac{702}{349}a-\frac{1351}{349}$, $\frac{719}{349}a^{10}-\frac{889}{349}a^{9}-\frac{2543}{349}a^{8}+\frac{5560}{349}a^{7}+\frac{1076}{349}a^{6}-\frac{6041}{349}a^{5}-\frac{1574}{349}a^{4}+\frac{1001}{349}a^{3}+\frac{5094}{349}a^{2}-\frac{63}{349}a-\frac{1647}{349}$ | 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: | \( 16.2407020956 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{3}\cdot 16.2407020956 \cdot 1}{2\cdot\sqrt{145065457639}}\cr\approx \mathstrut & 0.169231958945 \end{aligned}\]
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2389\) | $\Q_{2389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
\(2617\) | $\Q_{2617}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
\(23203\) | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |