Normalized defining polynomial
\( x^{16} - 7 x^{15} + 25 x^{14} - 28 x^{13} - 110 x^{12} + 648 x^{11} - 1621 x^{10} + 2390 x^{9} - 1337 x^{8} - 2440 x^{7} + 8238 x^{6} - 11873 x^{5} + 11601 x^{4} - 7806 x^{3} + 3700 x^{2} - 966 x + 225 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-267838446060935143677483=-\,3\cdot 17^{4}\cdot 41\cdot 97^{4}\cdot 131^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.12$ | 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: | $3, 17, 41, 97, 131$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5168529044497072200189192} a^{15} + \frac{1008903779330672399248153}{2584264522248536100094596} a^{14} - \frac{779809686318536479530869}{5168529044497072200189192} a^{13} + \frac{1368826471311487266901151}{5168529044497072200189192} a^{12} - \frac{1118140269191081182079183}{5168529044497072200189192} a^{11} - \frac{433409966266318847354261}{1722843014832357400063064} a^{10} + \frac{501914327354588283998345}{1292132261124268050047298} a^{9} - \frac{340351659107055467756903}{2584264522248536100094596} a^{8} - \frac{670551433160321829915359}{5168529044497072200189192} a^{7} + \frac{2141314390830786062739569}{5168529044497072200189192} a^{6} + \frac{439006630906024376822453}{1722843014832357400063064} a^{5} - \frac{752568935786196421300477}{2584264522248536100094596} a^{4} - \frac{195767304720096249675963}{1722843014832357400063064} a^{3} + \frac{319009652124079826140715}{1722843014832357400063064} a^{2} + \frac{777473236785500323165453}{5168529044497072200189192} a + \frac{404689609691767762188045}{1722843014832357400063064}$
Class group and class number
$C_{2}$, which has order $2$
Unit group
| Rank: | $8$ | 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: | \( 494158.632159 \) | 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 | ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | $x^{7} + x^{2} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| $17$ | 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.8.4.1 | $x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 41 | Data not computed | ||||||
| $97$ | 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 | Data not computed | ||||||