Normalized defining polynomial
\( x^{16} + 31 x^{14} + 391 x^{12} + 2571 x^{10} + 9351 x^{8} + 18270 x^{6} + 16840 x^{4} + 5200 x^{2} + 400 \)
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: | \(7984925229121000000000000=2^{12}\cdot 5^{12}\cdot 41^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.01$ | 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: | $2, 5, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{4} + \frac{1}{3} a$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{8} - \frac{1}{18} a^{6} + \frac{1}{18} a^{4} + \frac{7}{18} a^{2} + \frac{1}{9}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{9} - \frac{1}{18} a^{7} + \frac{1}{18} a^{5} + \frac{7}{18} a^{3} + \frac{1}{9} a$, $\frac{1}{180} a^{12} + \frac{1}{180} a^{10} - \frac{1}{20} a^{8} + \frac{11}{180} a^{6} - \frac{1}{2} a^{5} + \frac{17}{60} a^{4} + \frac{1}{18} a^{2} - \frac{1}{9}$, $\frac{1}{180} a^{13} + \frac{1}{180} a^{11} - \frac{1}{20} a^{9} + \frac{11}{180} a^{7} - \frac{13}{60} a^{5} - \frac{1}{2} a^{4} - \frac{4}{9} a^{3} - \frac{1}{2} a^{2} - \frac{1}{9} a$, $\frac{1}{360} a^{14} - \frac{1}{360} a^{12} - \frac{1}{360} a^{10} + \frac{19}{360} a^{8} + \frac{19}{360} a^{6} - \frac{1}{2} a^{5} + \frac{49}{180} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{1800} a^{15} + \frac{1}{1800} a^{13} + \frac{41}{1800} a^{11} + \frac{7}{600} a^{9} + \frac{1}{1800} a^{7} - \frac{4}{15} a^{5} - \frac{4}{15} a^{3} + \frac{2}{9} a$
Class group and class number
$C_{2}\times C_{8}$, which has order $16$ (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: | \( 79500.4741647 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2:D_4$ (as 16T43):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2:D_4$ |
| Character table for $C_2^2:D_4$ |
Intermediate fields
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 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 | R | ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |