Normalized defining polynomial
\( x^{20} + 410 x^{18} + 63140 x^{16} + 4936400 x^{14} + 217300000 x^{12} + 5561404000 x^{10} + 81710048000 x^{8} + 642948880000 x^{6} + 2252271040000 x^{4} + 2009131200000 x^{2} + 435715200000 \)
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: | \(3512039209944081509788983935664128000000000000000=2^{30}\cdot 5^{15}\cdot 41^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $267.47$ | 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: | $2, 5, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1640=2^{3}\cdot 5\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1640}(1,·)$, $\chi_{1640}(277,·)$, $\chi_{1640}(517,·)$, $\chi_{1640}(961,·)$, $\chi_{1640}(1609,·)$, $\chi_{1640}(1281,·)$, $\chi_{1640}(1357,·)$, $\chi_{1640}(1041,·)$, $\chi_{1640}(1173,·)$, $\chi_{1640}(1369,·)$, $\chi_{1640}(201,·)$, $\chi_{1640}(329,·)$, $\chi_{1640}(933,·)$, $\chi_{1640}(1253,·)$, $\chi_{1640}(529,·)$, $\chi_{1640}(1557,·)$, $\chi_{1640}(373,·)$, $\chi_{1640}(1289,·)$, $\chi_{1640}(573,·)$, $\chi_{1640}(597,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{20} a^{4}$, $\frac{1}{60} a^{5} - \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{120} a^{6} + \frac{1}{60} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{120} a^{7} - \frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{1200} a^{8} - \frac{1}{60} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{1200} a^{9} + \frac{1}{3} a$, $\frac{1}{295200} a^{10} + \frac{1}{3600} a^{8} - \frac{1}{360} a^{6} - \frac{1}{90} a^{4} - \frac{1}{18} a^{2}$, $\frac{1}{295200} a^{11} + \frac{1}{3600} a^{9} - \frac{1}{360} a^{7} + \frac{1}{180} a^{5} - \frac{2}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{2952000} a^{12} + \frac{1}{3600} a^{8} + \frac{1}{180} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{2952000} a^{13} + \frac{1}{3600} a^{9} + \frac{1}{180} a^{5} - \frac{1}{9} a^{3}$, $\frac{1}{5904000} a^{14} + \frac{1}{3600} a^{8} - \frac{1}{60} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{17712000} a^{15} + \frac{1}{2700} a^{9} + \frac{1}{180} a^{5} + \frac{4}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{38789280000} a^{16} + \frac{19}{258595200} a^{14} + \frac{1}{7183200} a^{12} - \frac{91}{96973200} a^{10} + \frac{41}{131400} a^{8} - \frac{22}{9855} a^{6} - \frac{1537}{118260} a^{4} - \frac{116}{657} a^{2} + \frac{1}{73}$, $\frac{1}{38789280000} a^{17} + \frac{11}{646488000} a^{15} + \frac{1}{7183200} a^{13} - \frac{91}{96973200} a^{11} - \frac{23}{394200} a^{9} - \frac{22}{9855} a^{7} - \frac{223}{118260} a^{5} + \frac{17}{1971} a^{3} - \frac{70}{219} a$, $\frac{1}{248860167950024640000} a^{18} - \frac{68135969}{62215041987506160000} a^{16} + \frac{285408786601}{4147669465833744000} a^{14} + \frac{808739005031}{6221504198750616000} a^{12} + \frac{816730619333}{622150419875061600} a^{10} + \frac{103374920039}{632266686864900} a^{8} - \frac{1652338088623}{758720024237880} a^{6} + \frac{1756362346433}{75872002423788} a^{4} - \frac{157675620271}{4215111245766} a^{2} + \frac{62530036437}{234172846987}$, $\frac{1}{248860167950024640000} a^{19} - \frac{68135969}{62215041987506160000} a^{17} + \frac{8539323269}{691278244305624000} a^{15} + \frac{808739005031}{6221504198750616000} a^{13} + \frac{816730619333}{622150419875061600} a^{11} - \frac{32699481737}{158066671716225} a^{9} - \frac{1652338088623}{758720024237880} a^{7} + \frac{351589240633}{379360012118940} a^{5} - \frac{119427006913}{6322666868649} a^{3} + \frac{62530036437}{234172846987} a$
Class group and class number
$C_{2}\times C_{10}\times C_{7752100}$, which has order $155042000$ (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: | \( 41023218.25673422 \) (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
| \(\Q(\sqrt{5}) \), 4.0.13448000.5, 5.5.2825761.1, 10.10.24952891341003125.1 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | $20$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | R | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $41$ | 41.10.9.1 | $x^{10} - 41$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 41.10.9.1 | $x^{10} - 41$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |