Normalized defining polynomial
\( x^{20} - 10 x^{19} + 54 x^{18} - 201 x^{17} + 564 x^{16} - 1248 x^{15} + 2230 x^{14} - 3256 x^{13} + 3901 x^{12} - 3828 x^{11} + 3061 x^{10} - 1990 x^{9} + 1065 x^{8} - 492 x^{7} + 193 x^{6} - 29 x^{5} - 58 x^{4} + 72 x^{3} - 38 x^{2} + 9 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3551123440117203753498174869=13^{10}\cdot 347^{4}\cdot 1776701\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.85$ | 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: | $13, 347, 1776701$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{5}{13} a^{15} - \frac{1}{13} a^{14} + \frac{4}{13} a^{13} - \frac{2}{13} a^{12} - \frac{1}{13} a^{11} - \frac{2}{13} a^{10} + \frac{4}{13} a^{9} + \frac{4}{13} a^{8} + \frac{1}{13} a^{7} - \frac{2}{13} a^{6} + \frac{3}{13} a^{5} + \frac{3}{13} a^{4} + \frac{2}{13} a^{2} - \frac{6}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{17} - \frac{4}{13} a^{14} + \frac{4}{13} a^{13} - \frac{4}{13} a^{12} + \frac{3}{13} a^{11} + \frac{1}{13} a^{10} - \frac{3}{13} a^{9} - \frac{6}{13} a^{8} + \frac{6}{13} a^{7} + \frac{1}{13} a^{5} - \frac{2}{13} a^{4} + \frac{2}{13} a^{3} - \frac{3}{13} a^{2} - \frac{5}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{18} - \frac{4}{13} a^{15} + \frac{4}{13} a^{14} - \frac{4}{13} a^{13} + \frac{3}{13} a^{12} + \frac{1}{13} a^{11} - \frac{3}{13} a^{10} - \frac{6}{13} a^{9} + \frac{6}{13} a^{8} + \frac{1}{13} a^{6} - \frac{2}{13} a^{5} + \frac{2}{13} a^{4} - \frac{3}{13} a^{3} - \frac{5}{13} a^{2} + \frac{6}{13} a$, $\frac{1}{13} a^{19} - \frac{2}{13} a^{15} + \frac{5}{13} a^{14} + \frac{6}{13} a^{13} + \frac{6}{13} a^{12} + \frac{6}{13} a^{11} - \frac{1}{13} a^{10} - \frac{4}{13} a^{9} + \frac{3}{13} a^{8} + \frac{5}{13} a^{7} + \frac{3}{13} a^{6} + \frac{1}{13} a^{5} - \frac{4}{13} a^{4} - \frac{5}{13} a^{3} + \frac{1}{13} a^{2} + \frac{2}{13} a + \frac{3}{13}$
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: | \( 605684.795832 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 245760 |
| The 201 conjugacy class representatives for t20n887 are not computed |
| Character table for t20n887 is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1 |
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 | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 347 | Data not computed | ||||||
| 1776701 | Data not computed | ||||||