Normalized defining polynomial
\( x^{20} + 43 x^{18} + 793 x^{16} + 8183 x^{14} + 51657 x^{12} + 204599 x^{10} + 500105 x^{8} + 707959 x^{6} + 500105 x^{4} + 119415 x^{2} + 4489 \)
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: | \(1646829531350307068613612031442944=2^{20}\cdot 7^{10}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $45.80$ | 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: | $2, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(308=2^{2}\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(195,·)$, $\chi_{308}(71,·)$, $\chi_{308}(139,·)$, $\chi_{308}(13,·)$, $\chi_{308}(15,·)$, $\chi_{308}(141,·)$, $\chi_{308}(83,·)$, $\chi_{308}(267,·)$, $\chi_{308}(153,·)$, $\chi_{308}(155,·)$, $\chi_{308}(225,·)$, $\chi_{308}(293,·)$, $\chi_{308}(295,·)$, $\chi_{308}(41,·)$, $\chi_{308}(167,·)$, $\chi_{308}(237,·)$, $\chi_{308}(113,·)$, $\chi_{308}(307,·)$, $\chi_{308}(169,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{67} a^{11} + \frac{22}{67} a^{9} - \frac{25}{67} a^{7} + \frac{13}{67} a^{5} + \frac{9}{67} a^{3} + \frac{17}{67} a$, $\frac{1}{1541} a^{12} - \frac{112}{1541} a^{10} + \frac{578}{1541} a^{8} + \frac{348}{1541} a^{6} - \frac{326}{1541} a^{4} + \frac{687}{1541} a^{2} + \frac{10}{23}$, $\frac{1}{1541} a^{13} + \frac{3}{1541} a^{11} + \frac{26}{1541} a^{9} + \frac{555}{1541} a^{7} - \frac{372}{1541} a^{5} + \frac{181}{1541} a^{3} - \frac{457}{1541} a$, $\frac{1}{1541} a^{14} + \frac{362}{1541} a^{10} + \frac{362}{1541} a^{8} + \frac{125}{1541} a^{6} - \frac{382}{1541} a^{4} + \frac{564}{1541} a^{2} - \frac{7}{23}$, $\frac{1}{1541} a^{15} - \frac{6}{1541} a^{11} - \frac{29}{1541} a^{9} + \frac{79}{1541} a^{7} - \frac{543}{1541} a^{5} + \frac{334}{1541} a^{3} - \frac{561}{1541} a$, $\frac{1}{1541} a^{16} - \frac{701}{1541} a^{10} + \frac{465}{1541} a^{8} + \frac{4}{1541} a^{6} - \frac{81}{1541} a^{4} + \frac{479}{1541} a^{2} - \frac{9}{23}$, $\frac{1}{1541} a^{17} - \frac{11}{1541} a^{11} + \frac{235}{1541} a^{9} - \frac{295}{1541} a^{7} - \frac{357}{1541} a^{5} + \frac{525}{1541} a^{3} + \frac{340}{1541} a$, $\frac{1}{1541} a^{18} + \frac{544}{1541} a^{10} - \frac{101}{1541} a^{8} + \frac{389}{1541} a^{6} + \frac{21}{1541} a^{4} + \frac{192}{1541} a^{2} - \frac{5}{23}$, $\frac{1}{1541} a^{19} - \frac{8}{1541} a^{11} + \frac{83}{1541} a^{9} + \frac{320}{1541} a^{7} + \frac{550}{1541} a^{5} - \frac{153}{1541} a^{3} - \frac{473}{1541} a$
Class group and class number
$C_{124}$, which has order $124$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1}{67} a^{11} + \frac{22}{67} a^{9} + \frac{176}{67} a^{7} + \frac{616}{67} a^{5} + \frac{880}{67} a^{3} + \frac{352}{67} a \) (order $4$) | 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: | \( 1415140.16249 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-77}) \), \(\Q(i, \sqrt{77})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.0.219503494144.1, 10.0.40581147486860288.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | R | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 11 | Data not computed | ||||||