Normalized defining polynomial
\( x^{20} - 5 x^{19} - 2 x^{18} + 41 x^{17} - 7 x^{16} - 203 x^{15} + 110 x^{14} + 512 x^{13} - 313 x^{12} - 799 x^{11} + 548 x^{10} + 791 x^{9} - 684 x^{8} - 632 x^{7} + 531 x^{6} + 467 x^{5} - 136 x^{4} - 190 x^{3} - 42 x^{2} + 2 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1759014386056767111350986161=3^{8}\cdot 401^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.03$ | 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: | $3, 401$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{228745331121644781} a^{19} + \frac{9377042222834917}{76248443707214927} a^{18} + \frac{20300632777528183}{76248443707214927} a^{17} - \frac{84862789389345553}{228745331121644781} a^{16} - \frac{20677093224099292}{76248443707214927} a^{15} + \frac{12321060581475767}{228745331121644781} a^{14} - \frac{16904598966219917}{76248443707214927} a^{13} + \frac{5829479952271025}{76248443707214927} a^{12} + \frac{43490665081264724}{228745331121644781} a^{11} - \frac{15030774680008733}{76248443707214927} a^{10} + \frac{9259369140021595}{76248443707214927} a^{9} - \frac{18328835307786991}{228745331121644781} a^{8} - \frac{4981956138849704}{228745331121644781} a^{7} - \frac{7043958052678595}{228745331121644781} a^{6} + \frac{93732716894546986}{228745331121644781} a^{5} + \frac{34650923903399066}{76248443707214927} a^{4} + \frac{72052500023871019}{228745331121644781} a^{3} + \frac{77531897142926467}{228745331121644781} a^{2} - \frac{54925504674829712}{228745331121644781} a - \frac{14586156363443523}{76248443707214927}$
Class group and class number
Trivial group, which has order $1$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1214073.35747 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4:D_5$ (as 20T38):
| A solvable group of order 160 |
| The 10 conjugacy class representatives for $C_2^4:D_5$ |
| Character table for $C_2^4:D_5$ |
Intermediate fields
| 5.5.160801.1, 10.6.2094413889681.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 401 | Data not computed | ||||||