Normalized defining polynomial
\( x^{16} - 8 x^{15} + 8 x^{14} + 84 x^{13} - 224 x^{12} - 112 x^{11} + 910 x^{10} - 568 x^{9} - 956 x^{8} + 852 x^{7} + 580 x^{6} - 452 x^{5} - 277 x^{4} + 80 x^{3} + 42 x^{2} + 40 x + 4 \)
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: | \(17983880206872670633984=2^{30}\cdot 7^{4}\cdot 17^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.60$ | 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: | $2, 7, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{14} a^{12} - \frac{3}{7} a^{11} + \frac{3}{14} a^{10} - \frac{1}{7} a^{9} + \frac{3}{14} a^{8} + \frac{2}{7} a^{7} - \frac{1}{14} a^{6} + \frac{3}{7} a^{5} - \frac{1}{2} a^{4} - \frac{2}{7} a^{3} + \frac{5}{14} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{14} a^{13} - \frac{5}{14} a^{11} + \frac{1}{7} a^{10} + \frac{5}{14} a^{9} - \frac{3}{7} a^{8} - \frac{5}{14} a^{7} + \frac{1}{14} a^{5} - \frac{2}{7} a^{4} - \frac{5}{14} a^{3} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{155974} a^{14} - \frac{1}{22282} a^{13} + \frac{5199}{155974} a^{12} - \frac{31103}{155974} a^{11} + \frac{10917}{22282} a^{10} + \frac{58823}{155974} a^{9} + \frac{64481}{155974} a^{8} - \frac{16865}{155974} a^{7} + \frac{835}{11998} a^{6} - \frac{5677}{22282} a^{5} + \frac{11671}{155974} a^{4} + \frac{46601}{155974} a^{3} + \frac{19148}{77987} a^{2} - \frac{34329}{77987} a - \frac{3275}{77987}$, $\frac{1}{198554902} a^{15} + \frac{629}{198554902} a^{14} - \frac{3414343}{99277451} a^{13} - \frac{985975}{99277451} a^{12} - \frac{26246526}{99277451} a^{11} - \frac{1602906}{5225129} a^{10} - \frac{46041390}{99277451} a^{9} + \frac{33893578}{99277451} a^{8} - \frac{5953686}{99277451} a^{7} + \frac{6601635}{99277451} a^{6} - \frac{1192823}{5225129} a^{5} + \frac{25933121}{99277451} a^{4} + \frac{76390745}{198554902} a^{3} - \frac{60751655}{198554902} a^{2} - \frac{46725513}{99277451} a + \frac{2395782}{99277451}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 263219.156169 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^5.C_2$ (as 16T511):
| A solvable group of order 256 |
| The 46 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
| \(\Q(\sqrt{17}) \), 4.4.9248.1, 4.2.1156.1, 4.2.2312.1, 8.4.134103990272.3, 8.8.134103990272.2, 8.4.85525504.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.2 | $x^{4} - 2 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.8.18.1 | $x^{8} + 14 x^{6} + 10 x^{4} + 12 x^{2} + 16 x + 4$ | $4$ | $2$ | $18$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ | |
| $7$ | 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |