Normalized defining polynomial
\( x^{16} + 20 x^{14} + 134 x^{12} + 335 x^{10} + 109 x^{8} - 650 x^{6} - 505 x^{4} + 250 x^{2} + 25 \)
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: | \(16141630445159040000000000=2^{16}\cdot 3^{12}\cdot 5^{10}\cdot 83^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.63$ | 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: | $2, 3, 5, 83$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{2}{5} a^{8} + \frac{1}{3} a^{6} - \frac{2}{5} a^{4} + \frac{1}{3}$, $\frac{1}{15} a^{13} - \frac{2}{5} a^{9} + \frac{1}{3} a^{7} - \frac{2}{5} a^{5} + \frac{1}{3} a$, $\frac{1}{2299425} a^{14} - \frac{484}{153295} a^{12} - \frac{2639}{17825} a^{10} - \frac{88427}{459885} a^{8} - \frac{303992}{766475} a^{6} - \frac{52553}{153295} a^{4} - \frac{4826}{10695} a^{2} + \frac{591}{30659}$, $\frac{1}{2299425} a^{15} - \frac{484}{153295} a^{13} - \frac{2639}{17825} a^{11} - \frac{88427}{459885} a^{9} - \frac{303992}{766475} a^{7} - \frac{52553}{153295} a^{5} - \frac{4826}{10695} a^{3} + \frac{591}{30659} a$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 863171.587527 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 6144 |
| The 78 conjugacy class representatives for t16n1665 are not computed |
| Character table for t16n1665 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.56025.1, 8.8.3138800625.1 |
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 | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.3 | $x^{4} + 2 x^{2} + 4 x + 4$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.12.12.3 | $x^{12} - 16 x^{10} - 51 x^{8} - 8 x^{6} + 43 x^{4} + 24 x^{2} - 57$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
| $3$ | 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.12.12.23 | $x^{12} + 21 x^{11} + 21 x^{10} + 63 x^{9} + 36 x^{8} + 54 x^{7} + 90 x^{6} + 81 x^{3} - 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
| $5$ | 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $83$ | 83.4.0.1 | $x^{4} - x + 22$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 83.4.0.1 | $x^{4} - x + 22$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 83.8.4.1 | $x^{8} + 303116 x^{4} - 571787 x^{2} + 22969827364$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |