Normalized defining polynomial
\( x^{24} - x^{23} + 2 x^{22} - 3 x^{21} + 5 x^{20} - 8 x^{19} + 13 x^{18} - 21 x^{17} + 34 x^{16} - 55 x^{15} + 89 x^{14} - 144 x^{13} + 233 x^{12} + 144 x^{11} + 89 x^{10} + 55 x^{9} + 34 x^{8} + 21 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1 \)
Invariants
| Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(784140351063197047157560791015625\)\(\medspace = 5^{12}\cdot 13^{22}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $23.47$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $5, 13$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $24$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(65=5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{65}(64,·)$, $\chi_{65}(1,·)$, $\chi_{65}(4,·)$, $\chi_{65}(6,·)$, $\chi_{65}(9,·)$, $\chi_{65}(11,·)$, $\chi_{65}(14,·)$, $\chi_{65}(16,·)$, $\chi_{65}(19,·)$, $\chi_{65}(21,·)$, $\chi_{65}(24,·)$, $\chi_{65}(29,·)$, $\chi_{65}(31,·)$, $\chi_{65}(34,·)$, $\chi_{65}(36,·)$, $\chi_{65}(41,·)$, $\chi_{65}(44,·)$, $\chi_{65}(46,·)$, $\chi_{65}(49,·)$, $\chi_{65}(51,·)$, $\chi_{65}(54,·)$, $\chi_{65}(56,·)$, $\chi_{65}(59,·)$, $\chi_{65}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{233} a^{13} - \frac{89}{233}$, $\frac{1}{233} a^{14} - \frac{89}{233} a$, $\frac{1}{233} a^{15} - \frac{89}{233} a^{2}$, $\frac{1}{233} a^{16} - \frac{89}{233} a^{3}$, $\frac{1}{233} a^{17} - \frac{89}{233} a^{4}$, $\frac{1}{233} a^{18} - \frac{89}{233} a^{5}$, $\frac{1}{233} a^{19} - \frac{89}{233} a^{6}$, $\frac{1}{233} a^{20} - \frac{89}{233} a^{7}$, $\frac{1}{233} a^{21} - \frac{89}{233} a^{8}$, $\frac{1}{233} a^{22} - \frac{89}{233} a^{9}$, $\frac{1}{233} a^{23} - \frac{89}{233} a^{10}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{5}{233} a^{18} - \frac{2584}{233} a^{5} \) (order $26$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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)];
| |
| Regulator: | \( 7346081.887826216 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2\times C_{12}$ (as 24T2):
| An abelian group of order 24 |
| The 24 conjugacy class representatives for $C_2\times C_{12}$ |
| Character table for $C_2\times C_{12}$ 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 13 | Data not computed | ||||||