Normalized defining polynomial
\( x^{11} - x^{10} - 5x^{9} + 3x^{8} + 10x^{7} - 3x^{6} - 11x^{5} + 3x^{4} + 5x^{3} - 2x^{2} + 1 \)
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: | \(64283038681\) \(\medspace = 71^{2}\cdot 3571^{2}\) | 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: | \(9.61\) | 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: | $71^{2/3}3571^{2/3}\approx 4005.8879678934163$ | ||
Ramified primes: | \(71\), \(3571\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}+\frac{1}{7}a^{9}-\frac{3}{7}a^{8}-\frac{3}{7}a^{7}-\frac{3}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{2}{7}a+\frac{3}{7}$
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: | $a$, $\frac{10}{7}a^{10}-\frac{18}{7}a^{9}-\frac{37}{7}a^{8}+\frac{61}{7}a^{7}+\frac{54}{7}a^{6}-\frac{76}{7}a^{5}-\frac{45}{7}a^{4}+\frac{73}{7}a^{3}-2a^{2}-\frac{20}{7}a+\frac{16}{7}$, $a^{10}-a^{9}-4a^{8}+2a^{7}+6a^{6}-a^{5}-5a^{4}+2a^{3}$, $\frac{31}{7}a^{10}-\frac{53}{7}a^{9}-\frac{114}{7}a^{8}+\frac{173}{7}a^{7}+\frac{173}{7}a^{6}-\frac{216}{7}a^{5}-\frac{171}{7}a^{4}+\frac{213}{7}a^{3}-a^{2}-\frac{48}{7}a+\frac{37}{7}$, $\frac{8}{7}a^{10}-\frac{13}{7}a^{9}-\frac{31}{7}a^{8}+\frac{46}{7}a^{7}+\frac{46}{7}a^{6}-\frac{65}{7}a^{5}-\frac{43}{7}a^{4}+\frac{71}{7}a^{3}-\frac{30}{7}a+\frac{10}{7}$, $\frac{6}{7}a^{10}-\frac{15}{7}a^{9}-\frac{18}{7}a^{8}+\frac{52}{7}a^{7}+\frac{24}{7}a^{6}-\frac{68}{7}a^{5}-\frac{20}{7}a^{4}+\frac{69}{7}a^{3}-3a^{2}-\frac{19}{7}a+\frac{18}{7}$ | 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: | \( 8.04640012169 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 8.04640012169 \cdot 1}{2\cdot\sqrt{64283038681}}\cr\approx \mathstrut & 0.197848558635 \end{aligned}\]
Galois group
A non-solvable group of order 19958400 |
The 31 conjugacy class representatives for $A_{11}$ |
Character table for $A_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
71.3.2.1 | $x^{3} + 71$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
71.4.0.1 | $x^{4} + 4 x^{2} + 41 x + 7$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(3571\) | $\Q_{3571}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |