Normalized defining polynomial
\( x^{16} + 3 x^{14} + 5 x^{12} + 7 x^{10} + 9 x^{8} - 4 x^{4} + 4 x^{2} + 4 \)
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: | \(81339259033747456=2^{24}\cdot 7^{8}\cdot 29^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $11.40$ | 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: | $2, 7, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8} a^{12} - \frac{3}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{3}{8} a^{11} - \frac{1}{2} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{376} a^{14} + \frac{9}{376} a^{12} - \frac{1}{2} a^{11} - \frac{35}{376} a^{10} - \frac{1}{2} a^{9} + \frac{79}{376} a^{8} - \frac{1}{2} a^{7} + \frac{13}{376} a^{6} - \frac{1}{2} a^{5} + \frac{43}{94} a^{4} - \frac{1}{2} a^{3} + \frac{91}{188} a^{2} - \frac{4}{47}$, $\frac{1}{752} a^{15} + \frac{9}{752} a^{13} - \frac{35}{752} a^{11} - \frac{1}{2} a^{10} + \frac{79}{752} a^{9} - \frac{363}{752} a^{7} - \frac{51}{188} a^{5} + \frac{91}{376} a^{3} + \frac{43}{94} a - \frac{1}{2}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{43}{376} a^{15} - \frac{105}{376} a^{13} - \frac{93}{376} a^{11} - \frac{107}{376} a^{9} - \frac{89}{376} a^{7} + \frac{203}{188} a^{5} + \frac{223}{188} a^{3} - \frac{79}{94} a \) (order $4$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 114.44554744 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times D_4^2.C_2$ (as 16T509):
| A solvable group of order 256 |
| The 40 conjugacy class representatives for $C_2\times D_4^2.C_2$ |
| Character table for $C_2\times D_4^2.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-1}) \), 4.0.392.1, 4.0.1568.1, \(\Q(i, \sqrt{7})\), 8.0.17825024.1, 8.0.71300096.1, 8.0.9834496.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |