Normalized defining polynomial
\(x^{24} - 2 x^{22} + 8 x^{18} - 16 x^{16} + 64 x^{12} - 256 x^{8} + 512 x^{6} - 2048 x^{2} + 4096\)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(2914041287899137980901233132568576\)\(\medspace = 2^{36}\cdot 3^{12}\cdot 7^{20}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $24.79$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $2, 3, 7$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Gal(K/\Q)|$: | $24$ | ||
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(168=2^{3}\cdot 3\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(5,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(13,·)$, $\chi_{168}(17,·)$, $\chi_{168}(149,·)$, $\chi_{168}(89,·)$, $\chi_{168}(25,·)$, $\chi_{168}(157,·)$, $\chi_{168}(101,·)$, $\chi_{168}(97,·)$, $\chi_{168}(37,·)$, $\chi_{168}(145,·)$, $\chi_{168}(41,·)$, $\chi_{168}(125,·)$, $\chi_{168}(109,·)$, $\chi_{168}(29,·)$, $\chi_{168}(113,·)$, $\chi_{168}(53,·)$, $\chi_{168}(73,·)$, $\chi_{168}(121,·)$, $\chi_{168}(61,·)$, $\chi_{168}(85,·)$$\rbrace$ | ||
This is a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$
Class group and class number
$C_{6}$, which has order $6$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -\frac{1}{2048} a^{22} \) (order $42$) ![]() | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 25576950.522236273 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_2^2\times C_6$ (as 24T3):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2^2\times C_6$ |
Character table for $C_2^2\times C_6$ is not computed |
Intermediate fields
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 | R | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.12.18.23 | $x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
2.12.18.23 | $x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
$3$ | 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
$7$ | 7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |