Normalized defining polynomial
\( x^{16} + 5 x^{14} + 233 x^{12} - 1005 x^{10} + 6655 x^{8} - 3685 x^{6} + 10928 x^{4} + 11520 x^{2} + 5184 \)
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: | \(3731986953978689760254088841=17^{14}\cdot 53^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $52.87$ | 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: | $17, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{24} a^{9} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{24} a^{10} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{1}{16} a^{4} - \frac{23}{48} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{912} a^{12} + \frac{1}{76} a^{10} - \frac{1}{48} a^{9} - \frac{5}{76} a^{8} - \frac{1}{4} a^{7} - \frac{7}{304} a^{6} - \frac{1}{4} a^{5} - \frac{7}{114} a^{4} - \frac{1}{16} a^{3} - \frac{11}{38} a^{2} + \frac{1}{6} a - \frac{15}{38}$, $\frac{1}{912} a^{13} - \frac{7}{912} a^{11} - \frac{1}{304} a^{9} - \frac{7}{304} a^{7} + \frac{115}{912} a^{5} + \frac{287}{912} a^{3} - \frac{1}{2} a^{2} - \frac{15}{38} a - \frac{1}{2}$, $\frac{1}{2833176049632} a^{14} + \frac{530235155}{2833176049632} a^{12} - \frac{10544806783}{2833176049632} a^{10} - \frac{1}{48} a^{9} - \frac{34442682671}{944392016544} a^{8} + \frac{70094319385}{2833176049632} a^{6} - \frac{1}{4} a^{5} + \frac{435991674539}{2833176049632} a^{4} - \frac{1}{16} a^{3} + \frac{128301875863}{354147006204} a^{2} + \frac{5}{12} a - \frac{3318350807}{19674833678}$, $\frac{1}{16999056297792} a^{15} - \frac{2576317531}{16999056297792} a^{13} - \frac{47823439015}{16999056297792} a^{11} + \frac{27688371049}{5666352099264} a^{9} - \frac{1}{8} a^{8} + \frac{2968507975423}{16999056297792} a^{7} - \frac{3639805449493}{16999056297792} a^{5} - \frac{123328891703}{2124882037224} a^{3} + \frac{1}{8} a^{2} + \frac{4020477431}{19674833678} a$
Class group and class number
$C_{506}$, which has order $506$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 128061.369114 \) (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
| \(\Q(\sqrt{17}) \), 4.4.260389.1, 4.4.4913.1, 4.4.15317.1, 8.0.61089990620221.2, 8.0.61089990620221.1, 8.8.67802431321.1 |
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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
| $53$ | 53.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 53.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.4.2.1 | $x^{4} + 477 x^{2} + 70225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 477 x^{2} + 70225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 477 x^{2} + 70225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |