Normalized defining polynomial
\( x^{16} - 8 x^{15} + 18 x^{14} + 14 x^{13} - 111 x^{12} + 120 x^{11} + 119 x^{10} - 408 x^{9} + 282 x^{8} + 352 x^{7} - 645 x^{6} + 90 x^{5} + 367 x^{4} - 222 x^{3} - 49 x^{2} + 80 x + 8 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(117629536214429777277659119616=2^{24}\cdot 13^{8}\cdot 41\cdot 47^{4}\cdot 42961\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.60$ | 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: | $2, 13, 41, 47, 42961$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{38} a^{14} - \frac{7}{38} a^{13} + \frac{17}{38} a^{12} - \frac{11}{38} a^{11} + \frac{9}{19} a^{10} - \frac{2}{19} a^{9} - \frac{5}{38} a^{8} - \frac{1}{2} a^{7} + \frac{5}{38} a^{6} + \frac{15}{38} a^{5} + \frac{4}{19} a^{4} - \frac{1}{19} a^{3} - \frac{5}{38} a^{2} - \frac{11}{38} a - \frac{7}{19}$, $\frac{1}{38} a^{15} + \frac{3}{19} a^{13} - \frac{3}{19} a^{12} + \frac{17}{38} a^{11} + \frac{4}{19} a^{10} + \frac{5}{38} a^{9} - \frac{8}{19} a^{8} - \frac{7}{19} a^{7} + \frac{6}{19} a^{6} - \frac{1}{38} a^{5} + \frac{8}{19} a^{4} - \frac{1}{2} a^{3} - \frac{4}{19} a^{2} - \frac{15}{38} a + \frac{8}{19}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 4104553182.83 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 24576 |
| The 88 conjugacy class representatives for t16n1806 are not computed |
| Character table for t16n1806 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 8.8.258421755904.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.1 | $x^{2} + 14$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.1 | $x^{2} + 14$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.12.18.23 | $x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
| $13$ | 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 41.3.0.1 | $x^{3} - x + 13$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 41.3.0.1 | $x^{3} - x + 13$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 41.6.0.1 | $x^{6} - x + 7$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $47$ | 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.4.0.1 | $x^{4} - x + 39$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 47.8.4.1 | $x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 42961 | Data not computed | ||||||