Normalized defining polynomial
\( x^{20} - 4 x^{19} - 35 x^{18} + 146 x^{17} + 455 x^{16} - 2044 x^{15} - 2696 x^{14} + 13960 x^{13} + 6764 x^{12} - 48836 x^{11} - 2270 x^{10} + 84688 x^{9} - 15516 x^{8} - 65992 x^{7} + 17949 x^{6} + 20748 x^{5} - 6419 x^{4} - 1814 x^{3} + 787 x^{2} - 72 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1470391355634309152000000000000000=2^{20}\cdot 5^{15}\cdot 11^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $45.54$ | 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: | $2, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(67,·)$, $\chi_{220}(69,·)$, $\chi_{220}(9,·)$, $\chi_{220}(203,·)$, $\chi_{220}(141,·)$, $\chi_{220}(207,·)$, $\chi_{220}(81,·)$, $\chi_{220}(3,·)$, $\chi_{220}(23,·)$, $\chi_{220}(89,·)$, $\chi_{220}(27,·)$, $\chi_{220}(163,·)$, $\chi_{220}(103,·)$, $\chi_{220}(169,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(147,·)$, $\chi_{220}(181,·)$, $\chi_{220}(201,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{884372941} a^{18} - \frac{433934556}{884372941} a^{17} + \frac{282192243}{884372941} a^{16} - \frac{394641417}{884372941} a^{15} + \frac{139771924}{884372941} a^{14} - \frac{322974840}{884372941} a^{13} + \frac{30938985}{884372941} a^{12} - \frac{151298292}{884372941} a^{11} - \frac{112545307}{884372941} a^{10} - \frac{215392746}{884372941} a^{9} + \frac{388285222}{884372941} a^{8} + \frac{64050219}{884372941} a^{7} + \frac{71148378}{884372941} a^{6} + \frac{174835562}{884372941} a^{5} + \frac{438006565}{884372941} a^{4} - \frac{263417180}{884372941} a^{3} + \frac{55954187}{884372941} a^{2} - \frac{704594}{884372941} a - \frac{82193344}{884372941}$, $\frac{1}{237956895637344157879439} a^{19} + \frac{132546192707036}{237956895637344157879439} a^{18} - \frac{23576606896499687087225}{237956895637344157879439} a^{17} + \frac{79078347160117178175112}{237956895637344157879439} a^{16} - \frac{21252820414485752588362}{237956895637344157879439} a^{15} - \frac{2789563297002485215964}{237956895637344157879439} a^{14} - \frac{88839568672317756725202}{237956895637344157879439} a^{13} - \frac{27400280445131474852716}{237956895637344157879439} a^{12} - \frac{90111022790693761137853}{237956895637344157879439} a^{11} - \frac{109735676887127840688256}{237956895637344157879439} a^{10} - \frac{7225421086519220050442}{237956895637344157879439} a^{9} - \frac{45908117503532501277987}{237956895637344157879439} a^{8} + \frac{52166771530386026883881}{237956895637344157879439} a^{7} - \frac{83420678356353414053256}{237956895637344157879439} a^{6} + \frac{41888525907520523954137}{237956895637344157879439} a^{5} - \frac{111861946957203324631331}{237956895637344157879439} a^{4} - \frac{47689514991000219004453}{237956895637344157879439} a^{3} + \frac{90040449429110135583898}{237956895637344157879439} a^{2} + \frac{21424294829096752494343}{237956895637344157879439} a - \frac{41887632459061143439478}{237956895637344157879439}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 8953413920.01 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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 | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | ||||||
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |