Normalized defining polynomial
\( x^{16} - x^{15} + 7 x^{14} + 10 x^{13} + 29 x^{12} + 38 x^{11} + 28 x^{10} + 28 x^{9} - 56 x^{8} + 55 x^{7} - 49 x^{6} - 60 x^{5} + 285 x^{4} - 460 x^{3} + 370 x^{2} - 150 x + 25 \)
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: | \(1512914374440000000000=2^{12}\cdot 3^{8}\cdot 5^{10}\cdot 7^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.07$ | 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: | $2, 3, 5, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{3}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{2}{5} a^{7} - \frac{1}{10} a^{6} + \frac{1}{5} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} - \frac{1}{10} a^{7} + \frac{1}{5} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{63358544145850} a^{15} - \frac{1055186468347}{31679272072925} a^{14} + \frac{1236363771627}{31679272072925} a^{13} - \frac{1332393698051}{31679272072925} a^{12} - \frac{333803874789}{12671708829170} a^{11} + \frac{10047502546304}{31679272072925} a^{10} - \frac{3290980776278}{31679272072925} a^{9} + \frac{7553949358691}{63358544145850} a^{8} - \frac{582817957759}{63358544145850} a^{7} + \frac{22024611386507}{63358544145850} a^{6} - \frac{2530189873074}{6335854414585} a^{5} + \frac{1130842961627}{6335854414585} a^{4} + \frac{814450397093}{2534341765834} a^{3} + \frac{4325197398883}{12671708829170} a^{2} - \frac{574673044857}{1267170882917} a + \frac{497580358691}{2534341765834}$
Class group and class number
$C_{2}$, which has order $2$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 13963.2452636 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3$ (as 16T217):
| A solvable group of order 128 |
| The 32 conjugacy class representatives for $C_2^4.C_2^3$ |
| Character table for $C_2^4.C_2^3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 8.2.9724050000.1, 8.2.9724050000.2, 8.4.1944810000.1 |
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 | R | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$ |
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 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.12.15 | $x^{8} + 2 x^{7} + 2 x^{4} + 12$ | $4$ | $2$ | $12$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |