Normalized defining polynomial
\( x^{20} - 7 x^{18} - 7 x^{17} + 36 x^{16} + 49 x^{15} - 98 x^{14} - 83 x^{13} + 266 x^{12} + 126 x^{11} - 364 x^{10} - 139 x^{9} + 229 x^{8} + 133 x^{7} - 108 x^{6} - 56 x^{5} + 35 x^{4} + 8 x^{3} - 10 x^{2} - x + 1 \)
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: | \(1342499736873772099501953125=5^{10}\cdot 29^{5}\cdot 1609^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.72$ | 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: | $5, 29, 1609$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1258439} a^{18} + \frac{384196}{1258439} a^{17} + \frac{268264}{1258439} a^{16} + \frac{16868}{1258439} a^{15} - \frac{209421}{1258439} a^{14} + \frac{126085}{1258439} a^{13} + \frac{38539}{1258439} a^{12} + \frac{49412}{179777} a^{11} - \frac{443743}{1258439} a^{10} + \frac{325662}{1258439} a^{9} - \frac{906}{96803} a^{8} + \frac{360345}{1258439} a^{7} - \frac{281586}{1258439} a^{6} + \frac{547}{1258439} a^{5} - \frac{460428}{1258439} a^{4} - \frac{71989}{1258439} a^{3} - \frac{85744}{179777} a^{2} + \frac{17186}{96803} a + \frac{321310}{1258439}$, $\frac{1}{30134745149509} a^{19} + \frac{8275759}{30134745149509} a^{18} + \frac{8151479937339}{30134745149509} a^{17} + \frac{13883167685066}{30134745149509} a^{16} - \frac{9734026150279}{30134745149509} a^{15} - \frac{127297399714}{2739522286319} a^{14} + \frac{11313621126820}{30134745149509} a^{13} - \frac{9155923910164}{30134745149509} a^{12} + \frac{954316496624}{30134745149509} a^{11} - \frac{5861837834858}{30134745149509} a^{10} - \frac{9816504446493}{30134745149509} a^{9} - \frac{13226788554215}{30134745149509} a^{8} + \frac{13251116462817}{30134745149509} a^{7} + \frac{3702088879297}{30134745149509} a^{6} - \frac{5593181435858}{30134745149509} a^{5} + \frac{14217981567793}{30134745149509} a^{4} + \frac{11098334219020}{30134745149509} a^{3} + \frac{9292537961923}{30134745149509} a^{2} + \frac{13531671607456}{30134745149509} a + \frac{1317146995146}{30134745149509}$
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: | \( 444214.312869 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 960 |
| The 35 conjugacy class representatives for t20n174 |
| Character table for t20n174 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 5.1.1609.1, 10.2.8090253125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | 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 | $20$ | $20$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | ||||||
| $29$ | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 1609 | Data not computed | ||||||