Normalized defining polynomial
\( x^{20} + 164 x^{18} + 10578 x^{16} + 338824 x^{14} + 5621264 x^{12} + 47086368 x^{10} + 203993368 x^{8} + 449537120 x^{6} + 458498736 x^{4} + 162171072 x^{2} + 8608032 \)
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: | \(158322645890088916737377685620382389712556392448=2^{55}\cdot 41^{19}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $229.07$ | 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: | $2, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(656=2^{4}\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{656}(1,·)$, $\chi_{656}(131,·)$, $\chi_{656}(537,·)$, $\chi_{656}(385,·)$, $\chi_{656}(651,·)$, $\chi_{656}(529,·)$, $\chi_{656}(595,·)$, $\chi_{656}(409,·)$, $\chi_{656}(25,·)$, $\chi_{656}(155,·)$, $\chi_{656}(419,·)$, $\chi_{656}(625,·)$, $\chi_{656}(105,·)$, $\chi_{656}(43,·)$, $\chi_{656}(305,·)$, $\chi_{656}(579,·)$, $\chi_{656}(531,·)$, $\chi_{656}(441,·)$, $\chi_{656}(635,·)$, $\chi_{656}(443,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{5} + \frac{1}{3} a$, $\frac{1}{18} a^{6} + \frac{1}{18} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{1}{9} a^{3}$, $\frac{1}{36} a^{8} + \frac{2}{9} a^{2}$, $\frac{1}{324} a^{9} + \frac{1}{54} a^{7} + \frac{1}{27} a^{5} - \frac{7}{81} a^{3} - \frac{2}{9} a$, $\frac{1}{324} a^{10} - \frac{1}{108} a^{8} - \frac{1}{54} a^{6} + \frac{2}{81} a^{4}$, $\frac{1}{324} a^{11} - \frac{1}{54} a^{7} + \frac{13}{162} a^{5} - \frac{4}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{1944} a^{12} + \frac{1}{972} a^{10} - \frac{1}{162} a^{8} + \frac{4}{243} a^{6} + \frac{37}{486} a^{4} + \frac{1}{27} a^{2}$, $\frac{1}{1944} a^{13} + \frac{1}{972} a^{11} - \frac{1}{486} a^{7} - \frac{35}{486} a^{5} - \frac{2}{81} a^{3} + \frac{2}{9} a$, $\frac{1}{5832} a^{14} + \frac{1}{5832} a^{12} + \frac{1}{1458} a^{10} + \frac{11}{1458} a^{8} + \frac{1}{729} a^{6} + \frac{49}{729} a^{4} + \frac{14}{81} a^{2}$, $\frac{1}{17496} a^{15} - \frac{1}{8748} a^{13} - \frac{1}{8748} a^{11} + \frac{13}{8748} a^{9} - \frac{11}{2187} a^{7} - \frac{337}{4374} a^{5} - \frac{4}{243} a^{3} + \frac{2}{9} a$, $\frac{1}{2554416} a^{16} + \frac{29}{1277208} a^{14} + \frac{145}{638604} a^{12} + \frac{38}{159651} a^{10} - \frac{2813}{319302} a^{8} - \frac{1177}{159651} a^{6} + \frac{2746}{53217} a^{4} + \frac{866}{5913} a^{2} - \frac{36}{73}$, $\frac{1}{2554416} a^{17} + \frac{29}{1277208} a^{15} + \frac{145}{638604} a^{13} + \frac{38}{159651} a^{11} + \frac{287}{638604} a^{9} - \frac{1177}{159651} a^{7} - \frac{3167}{53217} a^{5} - \frac{10}{5913} a^{3} - \frac{36}{73} a$, $\frac{1}{154721064320098992} a^{18} - \frac{4330702697}{51573688106699664} a^{16} - \frac{277512179849}{25786844053349832} a^{14} - \frac{3157007985851}{38680266080024748} a^{12} + \frac{19421259882173}{12893422026674916} a^{10} - \frac{27583516196039}{12893422026674916} a^{8} + \frac{411670887353963}{19340133040012374} a^{6} + \frac{85640012345146}{1074451835556243} a^{4} + \frac{14634796374019}{119383537284027} a^{2} - \frac{649746577081}{1473870830667}$, $\frac{1}{154721064320098992} a^{19} - \frac{4330702697}{51573688106699664} a^{17} - \frac{277512179849}{25786844053349832} a^{15} - \frac{3157007985851}{38680266080024748} a^{13} + \frac{19421259882173}{12893422026674916} a^{11} + \frac{6105498115985}{6446711013337458} a^{9} - \frac{304630336350199}{19340133040012374} a^{7} + \frac{131485512262283}{2148903671112486} a^{5} + \frac{17582538035353}{119383537284027} a^{3} + \frac{1489792206980}{4421612492001} a$
Class group and class number
$C_{2}\times C_{22}\times C_{14346200}$, which has order $631232800$ (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: | \( 6411717617.202166 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{82}) \), 4.0.141150208.4, 5.5.2825761.1, 10.10.10727651226221314048.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ | $20$ | $20$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | $20$ | R | $20$ | $20$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 41 | Data not computed | ||||||