Normalized defining polynomial
\( x^{16} - 4 x^{14} - 296 x^{12} + 2672 x^{10} - 2715 x^{8} - 23212 x^{6} + 17640 x^{4} + 63112 x^{2} + 9604 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1178591573237607342668775424=2^{46}\cdot 7^{4}\cdot 17^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $49.20$ | 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: | $2, 7, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{11} + \frac{3}{7} a^{9} - \frac{2}{7} a^{7} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{784} a^{12} - \frac{23}{392} a^{10} - \frac{15}{392} a^{8} - \frac{23}{98} a^{6} + \frac{113}{784} a^{4} + \frac{19}{56} a^{2} + \frac{1}{8}$, $\frac{1}{784} a^{13} - \frac{23}{392} a^{11} - \frac{15}{392} a^{9} - \frac{23}{98} a^{7} + \frac{113}{784} a^{5} + \frac{19}{56} a^{3} + \frac{1}{8} a$, $\frac{1}{89784081644768} a^{14} + \frac{7862799909}{22446020411192} a^{12} - \frac{1}{14} a^{11} + \frac{191771651707}{44892040822384} a^{10} + \frac{2}{7} a^{9} + \frac{4213165823807}{22446020411192} a^{8} + \frac{1}{7} a^{7} - \frac{249356425513}{1264564530208} a^{6} + \frac{1}{7} a^{5} - \frac{1424170032895}{3206574344456} a^{4} - \frac{1}{14} a^{3} - \frac{154271397197}{916164098416} a^{2} + \frac{23275317631}{65440292744}$, $\frac{1}{89784081644768} a^{15} + \frac{7862799909}{22446020411192} a^{13} + \frac{191771651707}{44892040822384} a^{11} + \frac{4213165823807}{22446020411192} a^{9} - \frac{249356425513}{1264564530208} a^{7} - \frac{1424170032895}{3206574344456} a^{5} - \frac{154271397197}{916164098416} a^{3} + \frac{23275317631}{65440292744} a$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 44424677.1952 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 64 |
| The 16 conjugacy class representatives for $D_4.D_4$ |
| Character table for $D_4.D_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.4.4352.1 x2, 4.4.9248.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.24.10 | $x^{8} + 16$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ |
| 2.8.22.90 | $x^{8} + 4 x^{7} + 14 x^{4} + 12 x^{2} + 2$ | $8$ | $1$ | $22$ | $D_4$ | $[2, 3, 7/2]$ | |
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |