Normalized defining polynomial
\( x^{11} - 3x^{10} + 9x^{8} - 6x^{7} - 11x^{6} + 6x^{5} + 13x^{4} - 4x^{3} - 8x^{2} + 2 \)
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: | \(303820448000\) \(\medspace = 2^{8}\cdot 5^{3}\cdot 9494389\) | 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: | \(11.06\) | 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: | $2^{2}5^{3/4}9494389^{1/2}\approx 41211.74513055835$ | ||
Ramified primes: | \(2\), \(5\), \(9494389\) | 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: | $\Q(\sqrt{47471945}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{71}a^{10}+\frac{19}{71}a^{9}-\frac{8}{71}a^{8}-\frac{25}{71}a^{7}+\frac{12}{71}a^{6}-\frac{31}{71}a^{5}+\frac{34}{71}a^{4}-\frac{20}{71}a^{3}-\frac{18}{71}a^{2}+\frac{22}{71}a-\frac{13}{71}$
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: | $\frac{7}{71}a^{10}-\frac{9}{71}a^{9}-\frac{56}{71}a^{8}+\frac{109}{71}a^{7}+\frac{84}{71}a^{6}-\frac{288}{71}a^{5}-\frac{46}{71}a^{4}+\frac{286}{71}a^{3}+\frac{158}{71}a^{2}-\frac{201}{71}a-\frac{91}{71}$, $\frac{44}{71}a^{10}-\frac{158}{71}a^{9}+\frac{74}{71}a^{8}+\frac{391}{71}a^{7}-\frac{466}{71}a^{6}-\frac{299}{71}a^{5}+\frac{431}{71}a^{4}+\frac{398}{71}a^{3}-\frac{295}{71}a^{2}-\frac{239}{71}a+\frac{67}{71}$, $\frac{13}{71}a^{10}-\frac{37}{71}a^{9}-\frac{33}{71}a^{8}+\frac{172}{71}a^{7}-\frac{57}{71}a^{6}-\frac{261}{71}a^{5}+\frac{87}{71}a^{4}+\frac{237}{71}a^{3}+\frac{50}{71}a^{2}-\frac{140}{71}a-\frac{27}{71}$, $\frac{74}{71}a^{10}-\frac{227}{71}a^{9}+\frac{47}{71}a^{8}+\frac{564}{71}a^{7}-\frac{461}{71}a^{6}-\frac{519}{71}a^{5}+\frac{244}{71}a^{4}+\frac{721}{71}a^{3}-\frac{196}{71}a^{2}-\frac{289}{71}a-\frac{39}{71}$, $\frac{81}{71}a^{10}-\frac{236}{71}a^{9}-\frac{9}{71}a^{8}+\frac{673}{71}a^{7}-\frac{377}{71}a^{6}-\frac{807}{71}a^{5}+\frac{198}{71}a^{4}+\frac{1007}{71}a^{3}-\frac{38}{71}a^{2}-\frac{490}{71}a-\frac{201}{71}$, $\frac{76}{71}a^{10}-\frac{189}{71}a^{9}-\frac{40}{71}a^{8}+\frac{514}{71}a^{7}-\frac{224}{71}a^{6}-\frac{510}{71}a^{5}-\frac{43}{71}a^{4}+\frac{539}{71}a^{3}+\frac{52}{71}a^{2}-\frac{174}{71}a-\frac{65}{71}$ | 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: | \( 34.9771751389 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 34.9771751389 \cdot 1}{2\cdot\sqrt{303820448000}}\cr\approx \mathstrut & 0.395599465828 \end{aligned}\]
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 | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(9494389\) | $\Q_{9494389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |