Normalized defining polynomial
\( x^{22} + 2x^{16} - x^{14} - x^{12} + x^{10} + x^{8} - 2x^{6} - x^{4} + 2x^{2} - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3115164990344181117349789696\) \(\medspace = 2^{22}\cdot 1559^{2}\cdot 17480927^{2}\) | 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: | \(17.77\) | 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: | not computed | ||
Ramified primes: | \(2\), \(1559\), \(17480927\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $a$, $2a^{20}-a^{18}+a^{16}+2a^{14}-2a^{12}-a^{10}+a^{8}+3a^{6}-4a^{4}+a^{2}+1$, $a^{20}+a^{18}+a^{16}+2a^{14}+a^{12}-a^{10}-a^{8}+a^{6}-a^{4}-a^{2}$, $2a^{18}-a^{16}+2a^{14}+2a^{12}-2a^{10}+2a^{4}-4a^{2}+2$, $a^{17}+a^{13}+a^{11}+2a^{3}-a$, $2a^{21}-3a^{19}+4a^{17}-3a^{13}+3a^{11}+a^{7}-6a^{5}+8a^{3}-4a$, $a^{21}-a^{20}+2a^{19}-a^{17}+3a^{15}-2a^{14}+a^{13}+a^{12}-3a^{11}+a^{10}-a^{8}+2a^{7}-a^{6}+a^{5}+2a^{4}-5a^{3}+a^{2}+3a-2$, $a^{20}+a^{18}+a^{16}+2a^{14}+a^{12}-a^{10}-a^{8}+a^{6}-a^{4}-a^{2}+a$, $a^{21}-2a^{20}+a^{19}+a^{17}-a^{16}+2a^{15}-3a^{14}+a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{7}-2a^{6}-a^{5}+3a^{4}-2a^{3}+a^{2}-1$, $3a^{21}+a^{20}+a^{19}+2a^{18}+2a^{17}-a^{16}+5a^{15}+3a^{14}+a^{12}-2a^{11}-3a^{10}+3a^{7}+2a^{6}-4a^{5}+a^{4}-2a^{3}-5a^{2}+a+3$, $a^{21}-3a^{20}+3a^{19}-a^{18}+a^{17}-2a^{16}+4a^{15}-5a^{14}+4a^{13}-a^{11}+2a^{10}-a^{9}+a^{7}-3a^{6}+a^{5}+4a^{4}-5a^{3}+3a^{2}-a$ (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)]
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Regulator: | \( 52464.2585693 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 52464.2585693 \cdot 1}{2\cdot\sqrt{3115164990344181117349789696}}\cr\approx \mathstrut & 0.180281719763 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.S_{11}$ (as 22T51):
A non-solvable group of order 40874803200 |
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$ are not computed |
Character table for $C_2^{10}.S_{11}$ is not computed |
Intermediate fields
11.3.27252765193.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 sibling: | data not computed |
Minimal sibling: | 22.2.25833082028433405007798095121937137823584835203516958882480351488906597451157612212005836699814723982224539479833814499328.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/31.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $22$ | $2$ | $11$ | $22$ | |||
\(1559\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(17480927\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |