Normalized defining polynomial
\( x^{16} + 33 x^{14} + 435 x^{12} + 3033 x^{10} + 12382 x^{8} + 30432 x^{6} + 43773 x^{4} + 33300 x^{2} + 10000 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2392244374201127641154089=13^{14}\cdot 157^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.39$ | 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: | $13, 157$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{102} a^{12} - \frac{4}{51} a^{10} + \frac{7}{102} a^{8} + \frac{13}{34} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{2}{51} a^{2} + \frac{13}{51}$, $\frac{1}{510} a^{13} - \frac{7}{85} a^{11} - \frac{1}{51} a^{9} - \frac{2}{85} a^{7} - \frac{7}{15} a^{5} - \frac{1}{2} a^{4} - \frac{41}{170} a^{3} - \frac{1}{2} a^{2} + \frac{43}{510} a$, $\frac{1}{520200} a^{14} - \frac{1}{1020} a^{13} + \frac{2333}{520200} a^{12} - \frac{43}{1020} a^{11} - \frac{1813}{104040} a^{10} - \frac{5}{68} a^{9} - \frac{38467}{520200} a^{8} - \frac{81}{340} a^{7} + \frac{116191}{260100} a^{6} + \frac{7}{30} a^{5} - \frac{32071}{65025} a^{4} - \frac{151}{510} a^{3} + \frac{17573}{520200} a^{2} - \frac{71}{340} a + \frac{1357}{5202}$, $\frac{1}{5202000} a^{15} + \frac{2333}{5202000} a^{13} + \frac{32867}{1040400} a^{11} - \frac{38467}{5202000} a^{9} - \frac{13859}{2601000} a^{7} - \frac{1}{2} a^{6} + \frac{81502}{325125} a^{5} - \frac{1}{2} a^{4} + \frac{364373}{5202000} a^{3} - \frac{2912}{13005} a - \frac{1}{2}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 787627.936673 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\wr C_4$ (as 16T158):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $C_2\wr C_4$ |
| Character table for $C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.4.344929.1, 4.0.2197.1, 4.0.26533.1, 8.4.1546688195533.1 x2, 8.0.9851517169.1 x2, 8.0.118976015041.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | 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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.8.7.2 | $x^{8} - 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 13.8.7.2 | $x^{8} - 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
| $157$ | 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 157.4.2.1 | $x^{4} + 1727 x^{2} + 887364$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 157.4.2.1 | $x^{4} + 1727 x^{2} + 887364$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |