Normalized defining polynomial
\( x^{16} - 4 x^{15} - 12 x^{14} + 60 x^{13} - 10 x^{12} - 92 x^{11} - 76 x^{10} + 52 x^{9} + 158 x^{8} + 52 x^{7} - 76 x^{6} - 92 x^{5} - 10 x^{4} + 60 x^{3} - 12 x^{2} - 4 x + 1 \)
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: | \(244231681894347217829888=2^{44}\cdot 17^{3}\cdot 41^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.96$ | 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: | $2, 17, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{53182} a^{14} - \frac{649}{26591} a^{13} + \frac{2183}{26591} a^{12} - \frac{5477}{26591} a^{11} - \frac{2903}{53182} a^{10} - \frac{4289}{26591} a^{9} - \frac{12279}{53182} a^{8} - \frac{1881}{26591} a^{7} - \frac{12279}{53182} a^{6} - \frac{4289}{26591} a^{5} + \frac{11844}{26591} a^{4} - \frac{5477}{26591} a^{3} - \frac{22225}{53182} a^{2} - \frac{649}{26591} a + \frac{1}{53182}$, $\frac{1}{53182} a^{15} - \frac{5205}{53182} a^{13} - \frac{7769}{53182} a^{12} + \frac{2495}{26591} a^{11} - \frac{375}{26591} a^{10} - \frac{2447}{26591} a^{9} + \frac{6348}{26591} a^{8} - \frac{2611}{53182} a^{7} + \frac{3940}{26591} a^{6} - \frac{22109}{53182} a^{5} + \frac{23465}{53182} a^{4} - \frac{7166}{26591} a^{3} - \frac{12352}{26591} a^{2} - \frac{4785}{26591} a + \frac{649}{26591}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 797506.714905 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 16384 |
| The 148 conjugacy class representatives for t16n1781 are not computed |
| Character table for t16n1781 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.29965156352.1 |
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 | $16$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | $16$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | $16$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | Data not computed | ||||||
| $17$ | 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.4.3.2 | $x^{4} - 153$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.8.0.1 | $x^{8} + x^{2} - 3 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $41$ | 41.8.0.1 | $x^{8} - x + 12$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 41.8.4.1 | $x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |