Normalized defining polynomial
\( x^{24} - 13 x^{20} + 143 x^{16} - 336 x^{12} + 663 x^{8} - 26 x^{4} + 1 \)
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: | \(4971225787269833056964369392336896\)\(\medspace = 2^{48}\cdot 3^{12}\cdot 7^{16}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $25.35$ | 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$$\chi_{168}(1,·)$, $\chi_{168}(67,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(11,·)$, $\chi_{168}(79,·)$, $\chi_{168}(107,·)$, $\chi_{168}(149,·)$, $\chi_{168}(23,·)$, $\chi_{168}(25,·)$, $\chi_{168}(71,·)$, $\chi_{168}(155,·)$, $\chi_{168}(29,·)$, $\chi_{168}(95,·)$, $\chi_{168}(163,·)$, $\chi_{168}(37,·)$, $\chi_{168}(43,·)$, $\chi_{168}(109,·)$, $\chi_{168}(113,·)$, $\chi_{168}(53,·)$, $\chi_{168}(121,·)$, $\chi_{168}(127,·)$, $\chi_{168}(151,·)$, $\chi_{168}(85,·)$$\rbrace$ | ||
This is a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{1}{13}$, $\frac{1}{13} a^{13} + \frac{1}{13} a$, $\frac{1}{13} a^{14} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} + \frac{1}{13} a^{3}$, $\frac{1}{377} a^{16} + \frac{10}{377} a^{12} - \frac{12}{29} a^{8} + \frac{66}{377} a^{4} - \frac{120}{377}$, $\frac{1}{377} a^{17} + \frac{10}{377} a^{13} - \frac{12}{29} a^{9} + \frac{66}{377} a^{5} - \frac{120}{377} a$, $\frac{1}{377} a^{18} + \frac{10}{377} a^{14} - \frac{12}{29} a^{10} + \frac{66}{377} a^{6} - \frac{120}{377} a^{2}$, $\frac{1}{377} a^{19} + \frac{10}{377} a^{15} - \frac{12}{29} a^{11} + \frac{66}{377} a^{7} - \frac{120}{377} a^{3}$, $\frac{1}{16211} a^{20} - \frac{11}{16211} a^{16} - \frac{395}{16211} a^{12} - \frac{51}{16211} a^{8} + \frac{2}{16211} a^{4} - \frac{1656}{16211}$, $\frac{1}{16211} a^{21} - \frac{11}{16211} a^{17} - \frac{395}{16211} a^{13} - \frac{51}{16211} a^{9} + \frac{2}{16211} a^{5} - \frac{1656}{16211} a$, $\frac{1}{16211} a^{22} - \frac{11}{16211} a^{18} - \frac{395}{16211} a^{14} - \frac{51}{16211} a^{10} + \frac{2}{16211} a^{6} - \frac{1656}{16211} a^{2}$, $\frac{1}{16211} a^{23} - \frac{11}{16211} a^{19} - \frac{395}{16211} a^{15} - \frac{51}{16211} a^{11} + \frac{2}{16211} a^{7} - \frac{1656}{16211} a^{3}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -\frac{7213}{16211} a^{23} + \frac{93791}{16211} a^{19} - \frac{1031701}{16211} a^{15} + \frac{2426101}{16211} a^{11} - \frac{4783341}{16211} a^{7} + \frac{187582}{16211} a^{3} \) (order $24$) | 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: | \( 24013423.00725467 \) (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 | Data not computed | ||||||
$3$ | 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 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
$7$ | 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |