Normalized defining polynomial
\( x^{16} - 2 x^{15} - 17 x^{14} + 20 x^{13} + 130 x^{12} - 96 x^{11} - 520 x^{10} + 344 x^{9} + 1135 x^{8} - 878 x^{7} - 1319 x^{6} + 1244 x^{5} + 692 x^{4} - 804 x^{3} - 118 x^{2} + 168 x + 4 \)
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: | \(506789311061954423422976=2^{41}\cdot 7^{7}\cdot 23^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.31$ | 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: | $2, 7, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{38} a^{14} + \frac{2}{19} a^{13} - \frac{3}{19} a^{12} + \frac{4}{19} a^{11} - \frac{5}{19} a^{10} + \frac{3}{19} a^{9} - \frac{6}{19} a^{8} + \frac{2}{19} a^{7} - \frac{15}{38} a^{6} + \frac{3}{19} a^{5} + \frac{7}{19} a^{4} - \frac{2}{19} a^{3} - \frac{4}{19} a^{2} - \frac{1}{19} a + \frac{6}{19}$, $\frac{1}{37719354009676} a^{15} - \frac{418675377145}{37719354009676} a^{14} - \frac{1753199762407}{18859677004838} a^{13} - \frac{602025537795}{9429838502419} a^{12} + \frac{996638965845}{9429838502419} a^{11} - \frac{16739635447}{9429838502419} a^{10} + \frac{4592727316550}{9429838502419} a^{9} + \frac{3756938285416}{9429838502419} a^{8} - \frac{12586097287109}{37719354009676} a^{7} + \frac{3694144746005}{37719354009676} a^{6} - \frac{8308118362877}{18859677004838} a^{5} - \frac{1355636884483}{9429838502419} a^{4} - \frac{738245542559}{18859677004838} a^{3} + \frac{7300692615419}{18859677004838} a^{2} + \frac{1189417587740}{9429838502419} a + \frac{2384485515726}{9429838502419}$
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: | \( 10509828.4683 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 16384 |
| The 136 conjugacy class representatives for t16n1770 are not computed |
| Character table for t16n1770 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.10304.1, 8.4.11891310592.2 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.1 | $x^{2} + 14$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.1 | $x^{2} + 14$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.4.11.15 | $x^{4} + 30$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
| 2.8.24.53 | $x^{8} + 8 x + 14$ | $8$ | $1$ | $24$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 7/2]^{2}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.8.7.1 | $x^{8} + 14$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ | |
| $23$ | 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.8.0.1 | $x^{8} + x^{2} - 2 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |