Normalized defining polynomial
\( x^{16} - 12 x^{14} + 46 x^{12} + 15 x^{10} + 229 x^{8} + 378 x^{6} - 1652 x^{4} - 3117 x^{2} + 16 \)
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: | \(7921483191352215537023449=17^{14}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.99$ | 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: | $17, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{64} a^{10} - \frac{1}{64} a^{8} - \frac{1}{8} a^{7} + \frac{11}{64} a^{4} - \frac{1}{2} a^{3} + \frac{5}{64} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{128} a^{11} - \frac{1}{128} a^{10} + \frac{7}{128} a^{9} - \frac{7}{128} a^{8} + \frac{11}{128} a^{5} - \frac{11}{128} a^{4} - \frac{35}{128} a^{3} - \frac{29}{128} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{44032} a^{12} + \frac{93}{22016} a^{10} - \frac{1755}{44032} a^{8} - \frac{2997}{44032} a^{6} + \frac{3943}{22016} a^{4} - \frac{12361}{44032} a^{2} + \frac{109}{2752}$, $\frac{1}{88064} a^{13} - \frac{1}{88064} a^{12} + \frac{93}{44032} a^{11} - \frac{93}{44032} a^{10} - \frac{1755}{88064} a^{9} + \frac{1755}{88064} a^{8} - \frac{2997}{88064} a^{7} + \frac{2997}{88064} a^{6} + \frac{3943}{44032} a^{5} - \frac{3943}{44032} a^{4} + \frac{31671}{88064} a^{3} + \frac{12361}{88064} a^{2} - \frac{2643}{5504} a - \frac{109}{5504}$, $\frac{1}{176128} a^{14} + \frac{1}{176128} a^{12} - \frac{389}{176128} a^{10} - \frac{153}{88064} a^{8} - \frac{10085}{176128} a^{6} - \frac{20967}{176128} a^{4} - \frac{20399}{176128} a^{2} + \frac{4603}{11008}$, $\frac{1}{352256} a^{15} - \frac{1}{352256} a^{14} + \frac{1}{352256} a^{13} - \frac{1}{352256} a^{12} - \frac{389}{352256} a^{11} + \frac{389}{352256} a^{10} - \frac{153}{176128} a^{9} + \frac{153}{176128} a^{8} - \frac{10085}{352256} a^{7} + \frac{10085}{352256} a^{6} - \frac{20967}{352256} a^{5} + \frac{20967}{352256} a^{4} + \frac{155729}{352256} a^{3} + \frac{20399}{352256} a^{2} + \frac{4603}{22016} a + \frac{6405}{22016}$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 5292206.5482 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $D_4:C_4$ |
| Character table for $D_4:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.2.93347.1, 4.4.4913.1, 4.2.5491.1, 8.2.2814512958107.1, 8.2.2814512958107.2, 8.4.8713662409.1 |
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 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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | R | R | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |