Normalized defining polynomial
\( x^{11} - 3x^{10} + 5x^{9} - 11x^{8} + 11x^{7} - 9x^{6} + 7x^{5} + x^{4} - 6x^{3} + 3x^{2} - 4x + 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1013364643329\) \(\medspace = 3^{4}\cdot 33829\cdot 369821\) | 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: | \(12.34\) | 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: | $3^{1/2}33829^{1/2}369821^{1/2}\approx 193731.83483103648$ | ||
Ramified primes: | \(3\), \(33829\), \(369821\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{12510674609}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{436}a^{10}-\frac{51}{109}a^{9}+\frac{25}{436}a^{8}+\frac{49}{109}a^{7}-\frac{145}{436}a^{6}-\frac{19}{109}a^{5}+\frac{23}{436}a^{4}+\frac{87}{218}a^{3}-\frac{25}{109}a^{2}+\frac{47}{436}a+\frac{141}{436}$
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)
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Fundamental units: | $a$, $\frac{37}{218}a^{10}-\frac{68}{109}a^{9}+\frac{271}{218}a^{8}-\frac{298}{109}a^{7}+\frac{739}{218}a^{6}-\frac{316}{109}a^{5}+\frac{633}{218}a^{4}+\frac{58}{109}a^{3}-\frac{215}{109}a^{2}+\frac{213}{218}a-\frac{451}{218}$, $\frac{8}{109}a^{10}+\frac{3}{109}a^{9}-\frac{18}{109}a^{8}-\frac{67}{109}a^{7}+\frac{39}{109}a^{6}-\frac{172}{109}a^{5}+\frac{293}{109}a^{4}-\frac{134}{109}a^{3}-\frac{37}{109}a^{2}+\frac{49}{109}a-\frac{71}{109}$, $\frac{73}{109}a^{10}-\frac{286}{109}a^{9}+\frac{517}{109}a^{8}-\frac{952}{109}a^{7}+\frac{1187}{109}a^{6}-\frac{752}{109}a^{5}+\frac{371}{109}a^{4}+\frac{167}{109}a^{3}-\frac{651}{109}a^{2}+\frac{488}{109}a-\frac{171}{109}$, $\frac{25}{218}a^{10}-\frac{43}{109}a^{9}+\frac{189}{218}a^{8}-\frac{166}{109}a^{7}+\frac{299}{218}a^{6}-\frac{187}{109}a^{5}-\frac{79}{218}a^{4}+\frac{104}{109}a^{3}-\frac{51}{109}a^{2}+\frac{521}{218}a-\frac{181}{218}$, $\frac{233}{218}a^{10}-\frac{331}{109}a^{9}+\frac{1029}{218}a^{8}-\frac{1146}{109}a^{7}+\frac{1967}{218}a^{6}-\frac{679}{109}a^{5}+\frac{999}{218}a^{4}+\frac{433}{109}a^{3}-\frac{641}{109}a^{2}+\frac{487}{218}a-\frac{719}{218}$ | 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: | \( 106.551564033 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 106.551564033 \cdot 1}{2\cdot\sqrt{1013364643329}}\cr\approx \mathstrut & 0.659867015419 \end{aligned}\]
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 | ${\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
\(33829\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(369821\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |