Normalized defining polynomial
\( x^{16} + 9 x^{14} - 843 x^{12} - 14469 x^{10} - 2679 x^{8} + 853160 x^{6} + 1398760 x^{4} - 11446600 x^{2} + 5856400 \)
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: | \(74998863177338890240000000000=2^{20}\cdot 5^{10}\cdot 11^{4}\cdot 29^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $63.78$ | 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: | $2, 5, 11, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{22} a^{10} + \frac{9}{22} a^{8} - \frac{7}{22} a^{6} + \frac{7}{22} a^{4} + \frac{5}{22} a^{2}$, $\frac{1}{22} a^{11} + \frac{9}{22} a^{9} - \frac{7}{22} a^{7} + \frac{7}{22} a^{5} + \frac{5}{22} a^{3}$, $\frac{1}{16940} a^{12} - \frac{101}{16940} a^{10} + \frac{5427}{16940} a^{8} - \frac{4019}{16940} a^{6} + \frac{6231}{16940} a^{4} + \frac{29}{154} a^{2} - \frac{3}{7}$, $\frac{1}{16940} a^{13} - \frac{101}{16940} a^{11} + \frac{5427}{16940} a^{9} - \frac{4019}{16940} a^{7} + \frac{6231}{16940} a^{5} + \frac{29}{154} a^{3} - \frac{3}{7} a$, $\frac{1}{2459592297585814206200} a^{14} - \frac{52304721511962141}{2459592297585814206200} a^{12} + \frac{49185607352717409127}{2459592297585814206200} a^{10} + \frac{432439473117199789061}{2459592297585814206200} a^{8} + \frac{546926157185538942311}{2459592297585814206200} a^{6} - \frac{4261613796982049317}{22359929978052856420} a^{4} + \frac{38283027753846142}{508180226773928555} a^{2} - \frac{1351621573857838}{9239640486798701}$, $\frac{1}{2459592297585814206200} a^{15} - \frac{52304721511962141}{2459592297585814206200} a^{13} + \frac{49185607352717409127}{2459592297585814206200} a^{11} + \frac{432439473117199789061}{2459592297585814206200} a^{9} + \frac{546926157185538942311}{2459592297585814206200} a^{7} - \frac{4261613796982049317}{22359929978052856420} a^{5} + \frac{38283027753846142}{508180226773928555} a^{3} - \frac{1351621573857838}{9239640486798701} a$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 132556422.384 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3$ (as 16T329):
| A solvable group of order 128 |
| The 29 conjugacy class representatives for $C_2^4.C_2^3$ |
| Character table for $C_2^4.C_2^3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.4 | $x^{8} + 2 x^{6} + 16$ | $2$ | $4$ | $12$ | $C_2^3: C_4$ | $[2, 2, 3]^{4}$ |
| 2.8.8.3 | $x^{8} + 2 x^{7} + 2 x^{6} + 16$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $29$ | 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |