Normalized defining polynomial
\( x^{16} - 17 x^{14} + 20 x^{12} + 201 x^{10} - 605 x^{8} - 395 x^{6} + 908 x^{4} - 477 x^{2} + 81 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1824496001102094673202161=13^{12}\cdot 23^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.83$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $13, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{26} a^{8} - \frac{1}{2} a^{7} - \frac{1}{13} a^{6} - \frac{1}{2} a^{5} - \frac{5}{26} a^{4} - \frac{1}{2} a^{3} + \frac{3}{13} a^{2} - \frac{1}{2} a + \frac{9}{26}$, $\frac{1}{78} a^{9} - \frac{5}{26} a^{7} - \frac{1}{2} a^{6} - \frac{3}{13} a^{5} - \frac{11}{26} a^{3} - \frac{1}{2} a^{2} + \frac{11}{39} a - \frac{1}{2}$, $\frac{1}{78} a^{10} + \frac{5}{13} a^{6} - \frac{1}{2} a^{5} - \frac{5}{13} a^{4} + \frac{17}{39} a^{2} - \frac{7}{26}$, $\frac{1}{78} a^{11} + \frac{5}{13} a^{7} - \frac{1}{2} a^{6} - \frac{5}{13} a^{5} + \frac{17}{39} a^{3} - \frac{7}{26} a$, $\frac{1}{78} a^{12} - \frac{1}{2} a^{7} + \frac{5}{13} a^{6} + \frac{14}{39} a^{4} + \frac{11}{26} a^{2} - \frac{6}{13}$, $\frac{1}{78} a^{13} - \frac{3}{26} a^{7} - \frac{11}{78} a^{5} - \frac{1}{2} a^{4} - \frac{1}{13} a^{3} + \frac{1}{26} a - \frac{1}{2}$, $\frac{1}{1193166} a^{14} + \frac{2240}{596583} a^{12} + \frac{431}{1193166} a^{10} - \frac{3268}{198861} a^{8} - \frac{1}{2} a^{7} - \frac{67061}{1193166} a^{6} - \frac{1}{2} a^{5} + \frac{343177}{1193166} a^{4} - \frac{1}{2} a^{3} - \frac{413581}{1193166} a^{2} - \frac{12108}{66287}$, $\frac{1}{3579498} a^{15} - \frac{10817}{3579498} a^{13} - \frac{7433}{1789749} a^{11} - \frac{3268}{596583} a^{9} + \frac{1401451}{3579498} a^{7} + \frac{373771}{3579498} a^{5} - \frac{1}{2} a^{4} - \frac{122657}{1789749} a^{3} + \frac{72665}{397722} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 752694.735162 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $D_4:C_4$ |
| Character table for $D_4:C_4$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.2.3887.1, 4.4.1162213.1, 4.2.50531.1, 8.2.58727785103.1, 8.2.347501687.1, 8.4.1350739057369.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 16 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $23$ | 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |