Normalized defining polynomial
\( x^{16} - 14 x^{14} + 87 x^{12} - 427 x^{10} + 2018 x^{8} - 6763 x^{6} + 12595 x^{4} - 10866 x^{2} + 2777 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(357845053850117818585088=2^{12}\cdot 23^{2}\cdot 2777^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.66$ | 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: | $2, 23, 2777$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{6}{13} a^{10} + \frac{3}{13} a^{8} - \frac{5}{13} a^{6} - \frac{2}{13} a^{4} + \frac{5}{13} a^{2} + \frac{6}{13}$, $\frac{1}{26} a^{13} - \frac{1}{26} a^{12} - \frac{3}{13} a^{11} + \frac{3}{13} a^{10} - \frac{5}{13} a^{9} + \frac{5}{13} a^{8} - \frac{5}{26} a^{7} + \frac{5}{26} a^{6} - \frac{1}{13} a^{5} + \frac{1}{13} a^{4} - \frac{4}{13} a^{3} + \frac{4}{13} a^{2} - \frac{7}{26} a + \frac{7}{26}$, $\frac{1}{24711388} a^{14} + \frac{581159}{24711388} a^{12} + \frac{284643}{12355694} a^{10} + \frac{6184251}{24711388} a^{8} + \frac{2294797}{24711388} a^{6} - \frac{4532311}{12355694} a^{4} + \frac{4611405}{24711388} a^{2} + \frac{723755}{1900876}$, $\frac{1}{49422776} a^{15} - \frac{1}{49422776} a^{14} + \frac{581159}{49422776} a^{13} - \frac{581159}{49422776} a^{12} - \frac{12071051}{24711388} a^{11} + \frac{12071051}{24711388} a^{10} - \frac{18527137}{49422776} a^{9} + \frac{18527137}{49422776} a^{8} + \frac{2294797}{49422776} a^{7} - \frac{2294797}{49422776} a^{6} + \frac{7823383}{24711388} a^{5} - \frac{7823383}{24711388} a^{4} - \frac{20099983}{49422776} a^{3} + \frac{20099983}{49422776} a^{2} - \frac{1177121}{3801752} a + \frac{1177121}{3801752}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 1391958.98709 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 12288 |
| The 64 conjugacy class representatives for t16n1764 are not computed |
| Character table for t16n1764 is not computed |
Intermediate fields
| 4.4.2777.1, 8.6.177369767.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | $16$ | $16$ | R | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.12.12.17 | $x^{12} + 22 x^{10} + 75 x^{8} - 12 x^{6} - 89 x^{4} + 54 x^{2} - 115$ | $2$ | $6$ | $12$ | 12T29 | $[2, 2]^{12}$ | |
| $23$ | 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2777 | Data not computed | ||||||