Normalized defining polynomial
\( x^{16} - 3 x^{14} - 15 x^{13} + 12 x^{12} + 86 x^{11} + 60 x^{10} - 462 x^{9} - 140 x^{8} + 375 x^{7} + 163 x^{6} - 507 x^{5} - 354 x^{4} + 1435 x^{3} + 2200 x^{2} + 1007 x + 131 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(8334464799024100000000=2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.45$ | 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: | $2, 5, 11, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} - \frac{2}{23} a^{12} + \frac{6}{23} a^{11} - \frac{3}{23} a^{10} + \frac{3}{23} a^{9} - \frac{7}{23} a^{8} + \frac{10}{23} a^{7} - \frac{1}{23} a^{6} - \frac{4}{23} a^{5} + \frac{5}{23} a^{4} + \frac{7}{23} a^{3} - \frac{11}{23} a^{2} + \frac{11}{23} a + \frac{9}{23}$, $\frac{1}{23} a^{14} + \frac{2}{23} a^{12} + \frac{9}{23} a^{11} - \frac{3}{23} a^{10} - \frac{1}{23} a^{9} - \frac{4}{23} a^{8} - \frac{4}{23} a^{7} - \frac{6}{23} a^{6} - \frac{3}{23} a^{5} - \frac{6}{23} a^{4} + \frac{3}{23} a^{3} - \frac{11}{23} a^{2} + \frac{8}{23} a - \frac{5}{23}$, $\frac{1}{976530711280452055937971} a^{15} + \frac{1578925254012298570154}{976530711280452055937971} a^{14} - \frac{20475718796650313370084}{976530711280452055937971} a^{13} - \frac{184141306874566285826002}{976530711280452055937971} a^{12} + \frac{156333565368912493470027}{976530711280452055937971} a^{11} + \frac{89303240060574064525714}{976530711280452055937971} a^{10} + \frac{93627146720535452027765}{976530711280452055937971} a^{9} + \frac{229628885176274755574851}{976530711280452055937971} a^{8} - \frac{469800705840871195939845}{976530711280452055937971} a^{7} - \frac{8785235135699089082736}{976530711280452055937971} a^{6} - \frac{11149661486090205232659}{42457857012193567649477} a^{5} - \frac{313334406867029729801745}{976530711280452055937971} a^{4} + \frac{262600215819154277686926}{976530711280452055937971} a^{3} - \frac{413297472267644674396298}{976530711280452055937971} a^{2} - \frac{274076424900748560081525}{976530711280452055937971} a - \frac{144348644565689810608464}{976530711280452055937971}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 27714.0253633 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^5.C_2$ (as 16T486):
| A solvable group of order 256 |
| The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$ |
| Character table for $C_2^2.C_2^5.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.475.1, 4.4.5225.1, 4.2.275.1, 8.4.27300625.1 |
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 | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 2.8.8.4 | $x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
| $5$ | 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 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $11$ | 11.4.2.2 | $x^{4} - 11 x^{2} + 847$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.2 | $x^{4} - 19 x^{2} + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |