Normalized defining polynomial
\( x^{16} - 6 x^{15} + 22 x^{14} - 56 x^{13} + 110 x^{12} - 174 x^{11} + 232 x^{10} - 266 x^{9} + 259 x^{8} - 202 x^{7} + 112 x^{6} - 30 x^{5} - 14 x^{4} + 20 x^{3} - 6 x^{2} - 2 x + 1 \)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(6884340550074368\)\(\medspace = 2^{24}\cdot 17^{7}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $9.77$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 17$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $8$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{298} a^{15} + \frac{31}{298} a^{14} - \frac{23}{298} a^{13} - \frac{13}{298} a^{12} - \frac{73}{298} a^{11} - \frac{22}{149} a^{10} - \frac{55}{298} a^{9} - \frac{33}{149} a^{8} - \frac{97}{298} a^{7} - \frac{33}{149} a^{6} - \frac{95}{298} a^{5} - \frac{59}{149} a^{4} + \frac{45}{149} a^{3} - \frac{77}{298} a^{2} - \frac{12}{149} a - \frac{145}{298}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{413}{2} a^{15} + 1123 a^{14} - 3912 a^{13} + \frac{18731}{2} a^{12} - \frac{34901}{2} a^{11} + \frac{52239}{2} a^{10} - \frac{66437}{2} a^{9} + 36242 a^{8} - \frac{66179}{2} a^{7} + \frac{46169}{2} a^{6} - \frac{20241}{2} a^{5} + 478 a^{4} + 3178 a^{3} - \frac{4703}{2} a^{2} - \frac{163}{2} a + \frac{737}{2} \) (order $4$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 23.5886655901 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_4.D_4:C_4$ (as 16T289):
| A solvable group of order 128 |
| The 44 conjugacy class representatives for $C_4.D_4:C_4$ |
| Character table for $C_4.D_4:C_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.0.272.1, 8.0.1257728.1 |
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 | $16$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | $16$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.8.7.2 | $x^{8} - 153$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |