Normalized defining polynomial
\( x^{20} - 3 x^{19} - x^{18} + 14 x^{17} - 21 x^{16} + 20 x^{15} - 6 x^{14} - 91 x^{13} + 224 x^{12} - 130 x^{11} - 184 x^{10} + 330 x^{9} - 224 x^{8} + 85 x^{7} + 27 x^{6} - 99 x^{5} + 81 x^{4} - 13 x^{3} - 16 x^{2} + 8 x - 1 \)
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: | \(11149939773041205947265625=5^{10}\cdot 97^{2}\cdot 3319^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.88$ | 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: | $5, 97, 3319$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{57} a^{17} + \frac{1}{57} a^{16} + \frac{22}{57} a^{15} + \frac{6}{19} a^{14} + \frac{5}{57} a^{13} - \frac{1}{19} a^{12} + \frac{28}{57} a^{11} - \frac{1}{3} a^{10} + \frac{20}{57} a^{9} + \frac{11}{57} a^{8} + \frac{4}{19} a^{7} - \frac{10}{57} a^{6} + \frac{3}{19} a^{5} - \frac{13}{57} a^{4} - \frac{2}{57} a^{3} - \frac{9}{19} a^{2} + \frac{20}{57} a + \frac{7}{57}$, $\frac{1}{57} a^{18} + \frac{2}{57} a^{16} - \frac{4}{57} a^{15} + \frac{25}{57} a^{14} + \frac{11}{57} a^{13} - \frac{26}{57} a^{12} + \frac{10}{57} a^{11} + \frac{20}{57} a^{10} - \frac{28}{57} a^{9} - \frac{6}{19} a^{8} + \frac{16}{57} a^{7} - \frac{1}{3} a^{6} - \frac{22}{57} a^{5} + \frac{11}{57} a^{4} - \frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{25}{57} a - \frac{26}{57}$, $\frac{1}{9805336707} a^{19} - \frac{80420005}{9805336707} a^{18} - \frac{14581244}{9805336707} a^{17} + \frac{1562594411}{9805336707} a^{16} - \frac{460940291}{9805336707} a^{15} - \frac{3938310281}{9805336707} a^{14} - \frac{380621266}{3268445569} a^{13} - \frac{669215217}{3268445569} a^{12} - \frac{1133177172}{3268445569} a^{11} + \frac{3553441687}{9805336707} a^{10} + \frac{1981420484}{9805336707} a^{9} + \frac{70410017}{516070353} a^{8} - \frac{1865314331}{9805336707} a^{7} - \frac{1281949136}{9805336707} a^{6} - \frac{459536672}{3268445569} a^{5} + \frac{2443166669}{9805336707} a^{4} + \frac{3910008680}{9805336707} a^{3} + \frac{3602786455}{9805336707} a^{2} - \frac{3743484014}{9805336707} a + \frac{4010606353}{9805336707}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 62751.6666339 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 102400 |
| The 130 conjugacy class representatives for t20n760 are not computed |
| Character table for t20n760 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.34424253125.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $97$ | 97.4.2.1 | $x^{4} + 873 x^{2} + 235225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 97.8.0.1 | $x^{8} - x + 84$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 97.8.0.1 | $x^{8} - x + 84$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 3319 | Data not computed | ||||||