magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, -100, -145, 825, 379, -2640, -450, 4290, 275, -4004, -90, 2275, 15, -800, -1, 170, 0, -20, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 + 170*x^16 - x^15 - 800*x^14 + 15*x^13 + 2275*x^12 - 90*x^11 - 4004*x^10 + 275*x^9 + 4290*x^8 - 450*x^7 - 2640*x^6 + 379*x^5 + 825*x^4 - 145*x^3 - 100*x^2 + 20*x + 1)
gp: K = bnfinit(x^20 - 20*x^18 + 170*x^16 - x^15 - 800*x^14 + 15*x^13 + 2275*x^12 - 90*x^11 - 4004*x^10 + 275*x^9 + 4290*x^8 - 450*x^7 - 2640*x^6 + 379*x^5 + 825*x^4 - 145*x^3 - 100*x^2 + 20*x + 1, 1)
Normalized defining polynomial
\( x^{20} - 20 x^{18} + 170 x^{16} - x^{15} - 800 x^{14} + 15 x^{13} + 2275 x^{12} - 90 x^{11} - 4004 x^{10} + 275 x^{9} + 4290 x^{8} - 450 x^{7} - 2640 x^{6} + 379 x^{5} + 825 x^{4} - 145 x^{3} - 100 x^{2} + 20 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
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: | \(171855208463966846466064453125=3^{10}\cdot 5^{35}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.96$ | 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: | $3, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(75=3\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{75}(64,·)$, $\chi_{75}(1,·)$, $\chi_{75}(2,·)$, $\chi_{75}(4,·)$, $\chi_{75}(8,·)$, $\chi_{75}(16,·)$, $\chi_{75}(17,·)$, $\chi_{75}(19,·)$, $\chi_{75}(23,·)$, $\chi_{75}(68,·)$, $\chi_{75}(31,·)$, $\chi_{75}(32,·)$, $\chi_{75}(34,·)$, $\chi_{75}(38,·)$, $\chi_{75}(46,·)$, $\chi_{75}(47,·)$, $\chi_{75}(49,·)$, $\chi_{75}(53,·)$, $\chi_{75}(61,·)$, $\chi_{75}(62,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.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: | \( 122630529.842 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.390625.1, \(\Q(\zeta_{25})^+\) |
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 | $20$ | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||