Normalized defining polynomial
\( x^{16} - 2 x^{15} + 15 x^{14} - 35 x^{13} + 113 x^{12} - 221 x^{11} + 535 x^{10} - 943 x^{9} + 1741 x^{8} - 2527 x^{7} + 3341 x^{6} - 3537 x^{5} + 3093 x^{4} - 2129 x^{3} + 1061 x^{2} - 332 x + 47 \)
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: | \(11740491199707618712081=17^{12}\cdot 67^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.95$ | 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: | $17, 67$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{26} a^{14} + \frac{9}{26} a^{13} - \frac{1}{26} a^{12} + \frac{11}{26} a^{11} + \frac{11}{26} a^{10} - \frac{1}{2} a^{9} + \frac{11}{26} a^{8} - \frac{3}{26} a^{7} + \frac{1}{26} a^{6} - \frac{1}{2} a^{5} - \frac{11}{26} a^{4} - \frac{5}{26} a^{3} - \frac{1}{2} a^{2} - \frac{7}{26} a + \frac{9}{26}$, $\frac{1}{43094255348943058} a^{15} + \frac{479154390600515}{43094255348943058} a^{14} - \frac{21346072825262993}{43094255348943058} a^{13} - \frac{2909856576517513}{43094255348943058} a^{12} + \frac{11243409560179255}{43094255348943058} a^{11} + \frac{12768586608877261}{43094255348943058} a^{10} - \frac{14823445642835403}{43094255348943058} a^{9} + \frac{6863122547883345}{43094255348943058} a^{8} + \frac{9672403090317445}{43094255348943058} a^{7} + \frac{1415300097558383}{43094255348943058} a^{6} - \frac{10107474073300953}{43094255348943058} a^{5} + \frac{5363794191136591}{43094255348943058} a^{4} + \frac{7600531318269125}{43094255348943058} a^{3} + \frac{8692322667514583}{43094255348943058} a^{2} + \frac{3199146936914517}{43094255348943058} a + \frac{1344413677802357}{21547127674471529}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 38350.1144855 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\wr C_4$ (as 16T157):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $C_2\wr C_4$ |
| Character table for $C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 8.2.1617217123.1 x2, 8.4.108353547241.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $17$ | 17.4.3.1 | $x^{4} - 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 17.4.3.1 | $x^{4} - 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.4.3.1 | $x^{4} - 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.4.3.1 | $x^{4} - 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $67$ | 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.4.2.1 | $x^{4} + 1541 x^{2} + 646416$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 67.4.2.1 | $x^{4} + 1541 x^{2} + 646416$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |