Normalized defining polynomial
\( x^{16} - 22 x^{14} + 324 x^{12} - 2607 x^{10} + 16368 x^{8} - 83974 x^{6} + 414546 x^{4} - 733381 x^{2} + 5285401 \)
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: | \(258168381614570521600000000=2^{16}\cdot 5^{8}\cdot 11^{8}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.74$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} + \frac{5}{11} a^{4}$, $\frac{1}{11} a^{9} + \frac{5}{11} a^{5}$, $\frac{1}{22} a^{10} - \frac{1}{22} a^{8} - \frac{3}{11} a^{6} - \frac{5}{22} a^{4} - \frac{1}{2}$, $\frac{1}{22} a^{11} - \frac{1}{22} a^{9} - \frac{3}{11} a^{7} - \frac{5}{22} a^{5} - \frac{1}{2} a$, $\frac{1}{484} a^{12} - \frac{17}{484} a^{8} - \frac{7}{44} a^{6} + \frac{13}{44} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{484} a^{13} - \frac{17}{484} a^{9} - \frac{7}{44} a^{7} + \frac{13}{44} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{1419050891747916152} a^{14} - \frac{80600457327021}{129004626522537832} a^{12} - \frac{6311650470934837}{1419050891747916152} a^{10} - \frac{2208674838214179}{64502313261268916} a^{8} + \frac{823469858125075}{64502313261268916} a^{6} - \frac{510710828334389}{1465961665028839} a^{4} - \frac{41996662022743}{143020650246716} a^{2} - \frac{24494710526079}{56113365168568}$, $\frac{1}{1419050891747916152} a^{15} - \frac{80600457327021}{129004626522537832} a^{13} - \frac{6311650470934837}{1419050891747916152} a^{11} - \frac{2208674838214179}{64502313261268916} a^{9} + \frac{823469858125075}{64502313261268916} a^{7} - \frac{510710828334389}{1465961665028839} a^{5} - \frac{41996662022743}{143020650246716} a^{3} - \frac{24494710526079}{56113365168568} a$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 1244815.01771 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 53 conjugacy class representatives for t16n868 are not computed |
| Character table for t16n868 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.475.1, 4.0.5225.1, 4.2.275.1, 8.0.27300625.1 |
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 | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.9 | $x^{8} + 6 x^{6} + 4 x^{5} + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.8.10 | $x^{8} + 2 x^{6} + 8 x^{3} + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
| $5$ | 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 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.8.6.1 | $x^{8} + 143 x^{4} + 5929$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
| $19$ | 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.8.4.1 | $x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |