Normalized defining polynomial
\( x^{16} - 8 x^{14} + 14 x^{12} + 12 x^{10} + 45 x^{8} + 92 x^{6} + 56 x^{4} + 8 x^{2} + 4 \)
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: | \(6338465731314712576=2^{40}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.97$ | 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: | $2, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{56968} a^{14} + \frac{1907}{56968} a^{12} - \frac{577}{28484} a^{10} - \frac{1}{8} a^{9} + \frac{4733}{56968} a^{8} + \frac{12949}{56968} a^{6} + \frac{1}{8} a^{5} - \frac{9629}{28484} a^{4} - \frac{1}{4} a^{3} - \frac{10359}{28484} a^{2} + \frac{1}{4} a + \frac{4373}{14242}$, $\frac{1}{113936} a^{15} - \frac{1}{113936} a^{14} + \frac{1907}{113936} a^{13} - \frac{1907}{113936} a^{12} + \frac{5967}{113936} a^{11} - \frac{5967}{113936} a^{10} + \frac{4733}{113936} a^{9} - \frac{4733}{113936} a^{8} + \frac{1457}{28484} a^{7} - \frac{1457}{28484} a^{6} + \frac{5867}{28484} a^{5} - \frac{5867}{28484} a^{4} + \frac{12623}{28484} a^{3} - \frac{12623}{28484} a^{2} + \frac{4373}{28484} a - \frac{4373}{28484}$
Class group and class number
$C_{2}$, which has order $2$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1084.24999819 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_8:C_2^2$ (as 16T35):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_8:C_2^2$ |
| Character table for $C_8:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \), 4.2.1792.1 x2, 4.0.1568.1 x2, \(\Q(\sqrt{2}, \sqrt{-7})\), 8.2.359661568.1 x2, 8.2.359661568.2 x2, 8.0.157351936.3 |
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 8 siblings: | data not computed |
| Degree 16 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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.20.58 | $x^{8} + 8 x^{6} + 64 x + 16$ | $8$ | $1$ | $20$ | $Q_8:C_2$ | $[2, 3, 3]^{2}$ |
| 2.8.20.58 | $x^{8} + 8 x^{6} + 64 x + 16$ | $8$ | $1$ | $20$ | $Q_8:C_2$ | $[2, 3, 3]^{2}$ | |
| $7$ | 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |