Normalized defining polynomial
\( x^{16} - 8 x^{14} - 27 x^{13} + 3 x^{12} + 113 x^{11} + 244 x^{10} + 342 x^{9} + 199 x^{8} + 14 x^{7} + 56 x^{6} + 102 x^{5} + 207 x^{4} + 74 x^{3} - 211 x^{2} - 129 x - 6 \)
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: | \(124120255491652871460297=3\cdot 17^{4}\cdot 19\cdot 97^{4}\cdot 131^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.76$ | 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: | $3, 17, 19, 97, 131$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1159112606262436483} a^{15} - \frac{357092119931346364}{1159112606262436483} a^{14} + \frac{275634899978274665}{1159112606262436483} a^{13} - \frac{28480267089802176}{165587515180348069} a^{12} - \frac{344693725389900471}{1159112606262436483} a^{11} - \frac{50822135269925204}{165587515180348069} a^{10} - \frac{570851065684271716}{1159112606262436483} a^{9} + \frac{412859517278534869}{1159112606262436483} a^{8} + \frac{198129093968130805}{1159112606262436483} a^{7} + \frac{467734338652098925}{1159112606262436483} a^{6} - \frac{421113194973518465}{1159112606262436483} a^{5} - \frac{381617018686676926}{1159112606262436483} a^{4} + \frac{370138237035733032}{1159112606262436483} a^{3} + \frac{181802760815614556}{1159112606262436483} a^{2} + \frac{33872766842491650}{1159112606262436483} a - \frac{76687884369100715}{1159112606262436483}$
Class group and class number
$C_{2}$, which has order $2$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 348162.519791 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 5160960 |
| The 106 conjugacy class representatives for t16n1944 are not computed |
| Character table for t16n1944 is not computed |
Intermediate fields
| 8.8.46664208361.1 |
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 | ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | R | R | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.7.0.1 | $x^{7} + x^{2} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 3.7.0.1 | $x^{7} + x^{2} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| $17$ | 17.8.0.1 | $x^{8} + x^{2} - 3 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 17.8.4.1 | $x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $19$ | 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.14.0.1 | $x^{14} + x^{2} - x + 15$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
| $97$ | 97.5.0.1 | $x^{5} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 97.5.0.1 | $x^{5} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 97.6.4.3 | $x^{6} + 873 x^{3} + 235225$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| $131$ | 131.2.1.2 | $x^{2} + 393$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 131.2.1.2 | $x^{2} + 393$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.4.2.1 | $x^{4} + 3537 x^{2} + 3363556$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 131.8.0.1 | $x^{8} - x + 56$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |