Normalized defining polynomial
\( x^{20} - 10 x^{19} + 52 x^{18} - 183 x^{17} + 484 x^{16} - 1016 x^{15} + 1751 x^{14} - 2535 x^{13} + 3136 x^{12} - 3359 x^{11} + 3147 x^{10} - 2596 x^{9} + 1888 x^{8} - 1203 x^{7} + 660 x^{6} - 302 x^{5} + 111 x^{4} - 32 x^{3} + 10 x^{2} - 4 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(255091649759845126953125=5^{10}\cdot 41\cdot 71^{2}\cdot 4861\cdot 5099^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.80$ | 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: | $5, 41, 71, 4861, 5099$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{59} a^{18} - \frac{9}{59} a^{17} + \frac{4}{59} a^{16} - \frac{5}{59} a^{15} + \frac{28}{59} a^{14} - \frac{26}{59} a^{13} - \frac{16}{59} a^{12} - \frac{3}{59} a^{11} - \frac{19}{59} a^{10} - \frac{16}{59} a^{9} - \frac{22}{59} a^{8} + \frac{12}{59} a^{7} - \frac{15}{59} a^{6} + \frac{25}{59} a^{5} - \frac{28}{59} a^{4} - \frac{7}{59} a^{3} + \frac{16}{59} a^{2} + \frac{21}{59} a - \frac{3}{59}$, $\frac{1}{59} a^{19} - \frac{18}{59} a^{17} - \frac{28}{59} a^{16} - \frac{17}{59} a^{15} - \frac{10}{59} a^{14} - \frac{14}{59} a^{13} - \frac{29}{59} a^{12} + \frac{13}{59} a^{11} - \frac{10}{59} a^{10} + \frac{11}{59} a^{9} - \frac{9}{59} a^{8} - \frac{25}{59} a^{7} + \frac{8}{59} a^{6} + \frac{20}{59} a^{5} - \frac{23}{59} a^{4} + \frac{12}{59} a^{3} - \frac{12}{59} a^{2} + \frac{9}{59} a - \frac{27}{59}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 1831.27226513 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 29491200 |
| The 702 conjugacy class representatives for t20n1045 are not computed |
| Character table for t20n1045 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.2.1131340625.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ | $20$ | R | $16{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | R | $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.12.6.1 | $x^{12} + 500 x^{6} - 3125 x^{2} + 62500$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 41 | Data not computed | ||||||
| 71 | Data not computed | ||||||
| 4861 | Data not computed | ||||||
| 5099 | Data not computed | ||||||