Normalized defining polynomial
\( x^{20} - 8 x^{17} - 7 x^{16} + 22 x^{15} + 14 x^{14} - 2 x^{13} - 81 x^{12} - 70 x^{11} + 116 x^{10} - 46 x^{9} - 21 x^{8} + 85 x^{7} - 10 x^{6} - 26 x^{5} + 11 x^{4} + 2 x^{3} - 7 x^{2} + x + 1 \)
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: | \(6288315747668637900390625=5^{10}\cdot 53^{2}\cdot 71^{2}\cdot 213247^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.38$ | 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: | $5, 53, 71, 213247$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} - \frac{2}{5} a^{16} + \frac{1}{5} a^{15} + \frac{2}{5} a^{14} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{1697510014876637075} a^{19} + \frac{153678154373500476}{1697510014876637075} a^{18} + \frac{634784742358352831}{1697510014876637075} a^{17} + \frac{679378454395630658}{1697510014876637075} a^{16} - \frac{190409944404852109}{1697510014876637075} a^{15} - \frac{252109417614557507}{1697510014876637075} a^{14} + \frac{618989127663940267}{1697510014876637075} a^{13} + \frac{41439397689236738}{339502002975327415} a^{12} + \frac{23911969148562444}{1697510014876637075} a^{11} - \frac{282810927436641841}{1697510014876637075} a^{10} - \frac{62263848772256944}{339502002975327415} a^{9} - \frac{205007742655928266}{1697510014876637075} a^{8} - \frac{316006571378883587}{1697510014876637075} a^{7} + \frac{451799359746011118}{1697510014876637075} a^{6} - \frac{751640610303971082}{1697510014876637075} a^{5} - \frac{615416126630163158}{1697510014876637075} a^{4} + \frac{249967124184528018}{1697510014876637075} a^{3} + \frac{69900418733669724}{339502002975327415} a^{2} - \frac{435976606798328472}{1697510014876637075} a - \frac{14803612461597031}{1697510014876637075}$
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: | \( 22636.8769621 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 7257600 |
| The 84 conjugacy class representatives for t20n1021 are not computed |
| Character table for t20n1021 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.2.802448461.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $53$ | 53.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 53.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.4.2.1 | $x^{4} + 477 x^{2} + 70225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.6.0.1 | $x^{6} - x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 53.6.0.1 | $x^{6} - x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $71$ | 71.2.1.2 | $x^{2} + 142$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 71.2.1.2 | $x^{2} + 142$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.8.0.1 | $x^{8} - 7 x + 13$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 71.8.0.1 | $x^{8} - 7 x + 13$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 213247 | Data not computed | ||||||