Normalized defining polynomial
\( x^{11} - 3x^{10} + x^{9} + 3x^{8} + 2x^{7} - 3x^{6} - 6x^{5} - 2x^{4} + 6x^{3} + 4x^{2} - x - 1 \)
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: | \(127663872125\) \(\medspace = 5^{3}\cdot 131\cdot 1297\cdot 6011\) | 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: | \(10.22\) | 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: | $5^{3/4}131^{1/2}1297^{1/2}6011^{1/2}\approx 106857.86753296934$ | ||
Ramified primes: | \(5\), \(131\), \(1297\), \(6011\) | 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: | $\Q(\sqrt{5106554885}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13}a^{10}+\frac{4}{13}a^{9}+\frac{3}{13}a^{8}-\frac{2}{13}a^{7}+\frac{1}{13}a^{6}+\frac{4}{13}a^{5}-\frac{4}{13}a^{4}-\frac{4}{13}a^{3}+\frac{4}{13}a^{2}+\frac{6}{13}a+\frac{2}{13}$
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: | $a^{10}-3a^{9}+a^{8}+3a^{7}+2a^{6}-3a^{5}-6a^{4}-2a^{3}+6a^{2}+4a-1$, $\frac{8}{13}a^{10}-\frac{33}{13}a^{9}+\frac{37}{13}a^{8}-\frac{3}{13}a^{7}+\frac{21}{13}a^{6}-\frac{46}{13}a^{5}-\frac{32}{13}a^{4}+\frac{7}{13}a^{3}+\frac{58}{13}a^{2}+\frac{9}{13}a-\frac{10}{13}$, $\frac{16}{13}a^{10}-\frac{53}{13}a^{9}+\frac{35}{13}a^{8}+\frac{33}{13}a^{7}+\frac{16}{13}a^{6}-\frac{40}{13}a^{5}-\frac{77}{13}a^{4}-\frac{12}{13}a^{3}+\frac{90}{13}a^{2}+\frac{31}{13}a-\frac{20}{13}$, $\frac{14}{13}a^{10}-\frac{48}{13}a^{9}+\frac{42}{13}a^{8}+\frac{11}{13}a^{7}+\frac{14}{13}a^{6}-\frac{35}{13}a^{5}-\frac{43}{13}a^{4}+\frac{9}{13}a^{3}+\frac{43}{13}a^{2}-\frac{7}{13}a-\frac{11}{13}$, $\frac{6}{13}a^{10}-\frac{28}{13}a^{9}+\frac{31}{13}a^{8}+\frac{14}{13}a^{7}-\frac{7}{13}a^{6}-\frac{41}{13}a^{5}-\frac{37}{13}a^{4}+\frac{41}{13}a^{3}+\frac{63}{13}a^{2}+\frac{10}{13}a-\frac{27}{13}$, $\frac{16}{13}a^{10}-\frac{53}{13}a^{9}+\frac{35}{13}a^{8}+\frac{33}{13}a^{7}+\frac{16}{13}a^{6}-\frac{40}{13}a^{5}-\frac{77}{13}a^{4}-\frac{12}{13}a^{3}+\frac{90}{13}a^{2}+\frac{18}{13}a-\frac{20}{13}$ | 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: | \( 13.1449090087 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 13.1449090087 \cdot 1}{2\cdot\sqrt{127663872125}}\cr\approx \mathstrut & 0.229352269843 \end{aligned}\]
Galois group
A non-solvable group of order 39916800 |
The 56 conjugacy class representatives for $S_{11}$ are not computed |
Character table for $S_{11}$ is not computed |
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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.7.0.1 | $x^{7} + 3 x + 3$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(131\) | 131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
131.9.0.1 | $x^{9} + 6 x^{3} + 6 x^{2} + 19 x + 129$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(1297\) | $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(6011\) | $\Q_{6011}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{6011}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |