Normalized defining polynomial
\( x^{16} - 2 x^{15} - 11 x^{14} + 17 x^{13} + 29 x^{12} - 43 x^{11} - 7 x^{10} + 35 x^{9} - 18 x^{8} + 35 x^{7} - 7 x^{6} - 43 x^{5} + 29 x^{4} + 17 x^{3} - 11 x^{2} - 2 x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-87101933678352892252840=-\,2^{3}\cdot 5\cdot 17^{4}\cdot 97^{4}\cdot 131^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.15$ | 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: | $2, 5, 17, 97, 131$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1873} a^{14} - \frac{704}{1873} a^{13} - \frac{276}{1873} a^{12} - \frac{319}{1873} a^{11} - \frac{517}{1873} a^{10} - \frac{152}{1873} a^{9} + \frac{453}{1873} a^{8} + \frac{591}{1873} a^{7} + \frac{453}{1873} a^{6} - \frac{152}{1873} a^{5} - \frac{517}{1873} a^{4} - \frac{319}{1873} a^{3} - \frac{276}{1873} a^{2} - \frac{704}{1873} a + \frac{1}{1873}$, $\frac{1}{3746} a^{15} - \frac{1}{3746} a^{14} - \frac{358}{1873} a^{13} + \frac{445}{3746} a^{12} - \frac{7}{1873} a^{11} - \frac{241}{3746} a^{10} + \frac{179}{1873} a^{9} - \frac{1233}{3746} a^{8} - \frac{1753}{3746} a^{7} + \frac{885}{1873} a^{6} + \frac{1261}{3746} a^{5} - \frac{204}{1873} a^{4} + \frac{227}{3746} a^{3} + \frac{30}{1873} a^{2} - \frac{439}{3746} a + \frac{703}{3746}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $12$ | 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: | \( 1099555.69048 \) | 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 | R | ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.7.0.1 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 2.7.0.1 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.14.0.1 | $x^{14} - x^{3} + x^{2} - 2 x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
| $17$ | 17.8.4.1 | $x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 17.8.0.1 | $x^{8} + x^{2} - 3 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $97$ | 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 | $x^{4} + 3537 x^{2} + 3363556$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 131.4.0.1 | $x^{4} - x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 131.4.0.1 | $x^{4} - x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |