Normalized defining polynomial
\( x^{16} - 4 x^{15} + 3 x^{14} + 21 x^{13} - 57 x^{12} + 15 x^{11} + 101 x^{10} - 180 x^{9} - 167 x^{8} + 122 x^{7} - 342 x^{6} + 101 x^{5} + 431 x^{4} + 212 x^{3} - 140 x^{2} - 32 x + 16 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(24575316877464616506889801=17^{12}\cdot 59^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $38.63$ | 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: | $17, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1628306120340911024} a^{15} + \frac{30897834037385563}{814153060170455512} a^{14} + \frac{365607081200353995}{1628306120340911024} a^{13} - \frac{105949705936954301}{1628306120340911024} a^{12} - \frac{281778048630927799}{1628306120340911024} a^{11} - \frac{109917236846398043}{1628306120340911024} a^{10} + \frac{107926104598456211}{1628306120340911024} a^{9} - \frac{19790712770475601}{814153060170455512} a^{8} + \frac{587726967897938617}{1628306120340911024} a^{7} - \frac{170954936421291909}{407076530085227756} a^{6} + \frac{66159377751943503}{814153060170455512} a^{5} - \frac{715390133036952663}{1628306120340911024} a^{4} + \frac{613480845533626633}{1628306120340911024} a^{3} + \frac{256038433130218753}{814153060170455512} a^{2} + \frac{182397614387448779}{407076530085227756} a - \frac{20669740986742001}{101769132521306939}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 2630288.39111 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4\wr C_2$ (as 16T28):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.2.17051.1, 4.4.4913.1, 4.2.289867.1, 8.4.84022877689.1 |
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 sibling: | 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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | R |
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 |
|---|---|---|---|---|---|---|---|
| 17 | Data not computed | ||||||
| $59$ | 59.4.2.2 | $x^{4} - 59 x^{2} + 6962$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 59.4.0.1 | $x^{4} - x + 14$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.8.4.1 | $x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |