Normalized defining polynomial
\( x^{11} - 11x^{9} - 216513 \)
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: | \(-3802786437878517362765933727\) \(\medspace = -\,3^{14}\cdot 11^{19}\cdot 13\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(321.57\) | 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: | $3^{235/108}11^{199/110}13^{1/2}\approx 3013.881853424313$ | ||
Ramified primes: | \(3\), \(11\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-143}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{5}-\frac{2}{27}a^{3}+\frac{1}{3}a$, $\frac{1}{81}a^{6}-\frac{2}{81}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{243}a^{7}-\frac{2}{243}a^{5}-\frac{2}{27}a^{3}+\frac{1}{3}a$, $\frac{1}{2187}a^{8}-\frac{11}{2187}a^{6}-\frac{1}{81}a^{5}+\frac{11}{81}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{6561}a^{9}-\frac{11}{6561}a^{7}-\frac{1}{243}a^{6}+\frac{11}{243}a^{4}+\frac{1}{9}a^{3}-\frac{2}{9}a$, $\frac{1}{19683}a^{10}-\frac{2}{19683}a^{8}-\frac{1}{729}a^{7}-\frac{11}{2187}a^{6}+\frac{2}{729}a^{5}+\frac{1}{27}a^{4}+\frac{11}{81}a^{3}-\frac{2}{27}a^{2}+\frac{1}{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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)
| |
Fundamental units: | $\frac{1}{19683}a^{10}-\frac{2}{6561}a^{9}+\frac{16}{19683}a^{8}-\frac{14}{6561}a^{7}+\frac{4}{729}a^{6}-\frac{10}{729}a^{5}+\frac{8}{243}a^{4}-\frac{2}{27}a^{3}+\frac{4}{27}a^{2}-\frac{2}{9}a-1$, $\frac{5051027}{19683}a^{10}-\frac{3619594}{6561}a^{9}-\frac{53323420}{19683}a^{8}+\frac{68082815}{6561}a^{7}-\frac{43980833}{2187}a^{6}-\frac{3869888}{729}a^{5}+\frac{46428274}{243}a^{4}-\frac{61406968}{81}a^{3}+\frac{42139391}{27}a^{2}+\frac{243803}{9}a-\frac{40315085}{3}$, $\frac{15\!\cdots\!83}{19683}a^{10}-\frac{15\!\cdots\!45}{6561}a^{9}-\frac{25\!\cdots\!69}{19683}a^{8}+\frac{31\!\cdots\!24}{6561}a^{7}+\frac{44\!\cdots\!84}{729}a^{6}-\frac{31\!\cdots\!77}{729}a^{5}-\frac{22\!\cdots\!62}{243}a^{4}+\frac{32\!\cdots\!79}{9}a^{3}-\frac{66\!\cdots\!20}{27}a^{2}-\frac{23\!\cdots\!38}{9}a+43\!\cdots\!35$, $\frac{27\!\cdots\!93}{6561}a^{10}-\frac{28\!\cdots\!61}{6561}a^{9}+\frac{63\!\cdots\!85}{6561}a^{8}+\frac{15\!\cdots\!54}{6561}a^{7}-\frac{39\!\cdots\!51}{2187}a^{6}+\frac{13\!\cdots\!06}{243}a^{5}-\frac{19\!\cdots\!01}{243}a^{4}-\frac{12\!\cdots\!56}{81}a^{3}+\frac{11\!\cdots\!74}{9}a^{2}-\frac{37\!\cdots\!66}{9}a+\frac{18\!\cdots\!61}{3}$, $\frac{25\!\cdots\!56}{6561}a^{10}+\frac{14\!\cdots\!03}{2187}a^{9}-\frac{49\!\cdots\!47}{6561}a^{8}+\frac{35\!\cdots\!39}{2187}a^{7}+\frac{12\!\cdots\!66}{2187}a^{6}-\frac{15\!\cdots\!58}{243}a^{5}-\frac{33\!\cdots\!76}{81}a^{4}+\frac{72\!\cdots\!32}{81}a^{3}+\frac{73\!\cdots\!16}{3}a^{2}-93\!\cdots\!91a-\frac{33\!\cdots\!85}{3}$ (assuming GRH) | 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: | \( 4462118626.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 4462118626.0 \cdot 1}{2\cdot\sqrt{3802786437878517362765933727}}\cr\approx \mathstrut & 0.70858097154 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/2.11.0.1}{11} }$ | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | R | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.6.10.9 | $x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 12$ | $6$ | $1$ | $10$ | $C_3^2:D_4$ | $[9/4, 9/4]_{4}^{2}$ | |
\(11\) | 11.11.19.3 | $x^{11} + 22 x^{9} + 11$ | $11$ | $1$ | $19$ | $F_{11}$ | $[19/10]_{10}$ |
\(13\) | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |