Normalized defining polynomial
\( x^{20} - 2 x^{19} - 11 x^{18} + 29 x^{17} + 11 x^{16} - 122 x^{15} + 184 x^{14} + 47 x^{13} - 342 x^{12} + 569 x^{11} - 536 x^{10} - 1288 x^{9} + 403 x^{8} + 3536 x^{7} + 167 x^{6} - 3112 x^{5} - 1429 x^{4} + 1221 x^{3} + 772 x^{2} - 92 x - 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-806877848402725521973046875=-\,5^{8}\cdot 151^{4}\cdot 331^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $22.15$ | 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: | $5, 151, 331$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{1}{11} a^{17} - \frac{2}{11} a^{16} - \frac{5}{11} a^{15} - \frac{3}{11} a^{14} + \frac{1}{11} a^{13} + \frac{1}{11} a^{12} + \frac{3}{11} a^{11} + \frac{1}{11} a^{10} - \frac{5}{11} a^{9} - \frac{3}{11} a^{8} + \frac{1}{11} a^{7} + \frac{4}{11} a^{5} - \frac{5}{11} a^{4} + \frac{3}{11} a^{3} - \frac{2}{11} a^{2} - \frac{5}{11} a - \frac{1}{11}$, $\frac{1}{55322426810941806093943306007} a^{19} - \frac{1895623486413689431942303564}{55322426810941806093943306007} a^{18} - \frac{6768513134738628772144109928}{55322426810941806093943306007} a^{17} - \frac{22444730375667593126739886560}{55322426810941806093943306007} a^{16} + \frac{7726611929152110381079938418}{55322426810941806093943306007} a^{15} + \frac{2670577102391803770440876844}{55322426810941806093943306007} a^{14} + \frac{1266511172719968812875722507}{55322426810941806093943306007} a^{13} + \frac{1464416928296828083627882198}{55322426810941806093943306007} a^{12} + \frac{387302298981036710267691641}{55322426810941806093943306007} a^{11} + \frac{20822289935149113031924920084}{55322426810941806093943306007} a^{10} - \frac{2503547159406850492764439955}{55322426810941806093943306007} a^{9} - \frac{25327165449396621971266810791}{55322426810941806093943306007} a^{8} + \frac{1753187917891559197030025743}{5029311528267436917631209637} a^{7} - \frac{833655307717794315276302303}{55322426810941806093943306007} a^{6} - \frac{14409178490200320799122315192}{55322426810941806093943306007} a^{5} + \frac{12321258342240768115128306266}{55322426810941806093943306007} a^{4} + \frac{4830098281789180520164176922}{55322426810941806093943306007} a^{3} + \frac{12319584702584538937055446856}{55322426810941806093943306007} a^{2} - \frac{6128880344951354856329001792}{55322426810941806093943306007} a - \frac{1628543448931749534169768882}{5029311528267436917631209637}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 1133873.36921 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 15000 |
| The 190 conjugacy class representatives for t20n462 are not computed |
| Character table for t20n462 is not computed |
Intermediate fields
| 4.2.331.1 |
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 30 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 | $20$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ | $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | $20$ | $15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ | $15{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{5}$ | $15{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
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 | ||||||
| 151 | Data not computed | ||||||
| 331 | Data not computed | ||||||