Normalized defining polynomial
\( x^{16} + 99 x^{14} + 3773 x^{12} + 73404 x^{10} + 804493 x^{8} + 5096286 x^{6} + 18172908 x^{4} + 33258186 x^{2} + 24019801 \)
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: | \(307604973831348243481600000000=2^{16}\cdot 5^{8}\cdot 13^{4}\cdot 29^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $69.66$ | 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: | $2, 5, 13, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{5}{13} a^{8} + \frac{3}{13} a^{6} + \frac{6}{13} a^{4} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{11} - \frac{5}{13} a^{9} + \frac{3}{13} a^{7} + \frac{6}{13} a^{5} + \frac{1}{13} a^{3}$, $\frac{1}{4901} a^{12} + \frac{99}{4901} a^{10} - \frac{1128}{4901} a^{8} - \frac{111}{4901} a^{6} + \frac{729}{4901} a^{4} - \frac{2}{13} a^{2}$, $\frac{1}{4901} a^{13} + \frac{99}{4901} a^{11} - \frac{1128}{4901} a^{9} - \frac{111}{4901} a^{7} + \frac{729}{4901} a^{5} - \frac{2}{13} a^{3}$, $\frac{1}{14231911888101193} a^{14} + \frac{643635756018}{14231911888101193} a^{12} + \frac{103162082237867}{14231911888101193} a^{10} - \frac{752818039291650}{14231911888101193} a^{8} - \frac{5500949333288031}{14231911888101193} a^{6} - \frac{16743926123311}{37750429411409} a^{4} - \frac{570392969235}{2903879185493} a^{2} + \frac{1229206197}{7702597309}$, $\frac{1}{14231911888101193} a^{15} + \frac{643635756018}{14231911888101193} a^{13} + \frac{103162082237867}{14231911888101193} a^{11} - \frac{752818039291650}{14231911888101193} a^{9} - \frac{5500949333288031}{14231911888101193} a^{7} - \frac{16743926123311}{37750429411409} a^{5} - \frac{570392969235}{2903879185493} a^{3} + \frac{1229206197}{7702597309} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2330}$, which has order $18640$ (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: | \( 3793.72993285 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$OD_{16}:C_2^2$ (as 16T106):
| A solvable group of order 64 |
| The 22 conjugacy class representatives for $OD_{16}:C_2^2$ |
| Character table for $OD_{16}:C_2^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.5 | $x^{8} + 2 x^{7} + 8 x^{2} + 16$ | $2$ | $4$ | $8$ | $C_8:C_2$ | $[2, 2]^{4}$ |
| 2.8.8.5 | $x^{8} + 2 x^{7} + 8 x^{2} + 16$ | $2$ | $4$ | $8$ | $C_8:C_2$ | $[2, 2]^{4}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $13$ | 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $29$ | 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |