Normalized defining polynomial
\( x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
Invariants
Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
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Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
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Discriminant: | \(-39471584120695485887249589623=-\,23^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
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Root discriminant: | $19.94$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
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Ramified primes: | $23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{23}(1,·)$, $\chi_{23}(2,·)$, $\chi_{23}(3,·)$, $\chi_{23}(4,·)$, $\chi_{23}(5,·)$, $\chi_{23}(6,·)$, $\chi_{23}(7,·)$, $\chi_{23}(8,·)$, $\chi_{23}(9,·)$, $\chi_{23}(10,·)$, $\chi_{23}(11,·)$, $\chi_{23}(12,·)$, $\chi_{23}(13,·)$, $\chi_{23}(14,·)$, $\chi_{23}(15,·)$, $\chi_{23}(16,·)$, $\chi_{23}(17,·)$, $\chi_{23}(18,·)$, $\chi_{23}(19,·)$, $\chi_{23}(20,·)$, $\chi_{23}(21,·)$, $\chi_{23}(22,·)$$\rbrace$ | ||
This is a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
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Torsion generator: | \( a \) (order $46$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
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Fundamental units: | \( a^{4} - a^{3} \), \( a^{4} - a \), \( a^{11} + a^{9} + a^{7} \), \( a^{20} - a^{3} \), \( a^{16} + a^{6} \), \( a^{11} + a^{9} \), \( a^{21} - a^{14} + a^{5} \), \( a^{11} - a^{8} + a^{5} \), \( a^{21} - a^{20} - a^{18} - a^{16} + a^{15} + a^{13} + a^{11} - a^{10} - a^{8} + a^{7} - a^{6} + a^{5} + a^{3} - a^{2} - 1 \), \( a^{16} + a^{8} + 1 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
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Regulator: | \( 1038656.82438 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
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Galois group
A cyclic group of order 22 |
The 22 conjugacy class representatives for $C_{22}$ |
Character table for $C_{22}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-23}) \), \(\Q(\zeta_{23})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | $22$ | $22$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
23 | Data not computed |