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Properties

Label 20.0.27861577025...0736.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 41^{19}$
Root discriminant $166.82$
Ramified primes $2, 3, 41$
Class number $17234432$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 134644]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2479113216, 0, 15287864832, 0, 22036561920, 0, 12636590976, 0, 3338003520, 0, 433908576, 0, 30021840, 0, 1142424, 0, 23616, 0, 246, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 246*x^18 + 23616*x^16 + 1142424*x^14 + 30021840*x^12 + 433908576*x^10 + 3338003520*x^8 + 12636590976*x^6 + 22036561920*x^4 + 15287864832*x^2 + 2479113216)
 
gp: K = bnfinit(x^20 + 246*x^18 + 23616*x^16 + 1142424*x^14 + 30021840*x^12 + 433908576*x^10 + 3338003520*x^8 + 12636590976*x^6 + 22036561920*x^4 + 15287864832*x^2 + 2479113216, 1)
 

Normalized defining polynomial

\( x^{20} + 246 x^{18} + 23616 x^{16} + 1142424 x^{14} + 30021840 x^{12} + 433908576 x^{10} + 3338003520 x^{8} + 12636590976 x^{6} + 22036561920 x^{4} + 15287864832 x^{2} + 2479113216 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(278615770255442274940771310275732270781300736=2^{30}\cdot 3^{10}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $166.82$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(984=2^{3}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{984}(1,·)$, $\chi_{984}(5,·)$, $\chi_{984}(961,·)$, $\chi_{984}(77,·)$, $\chi_{984}(941,·)$, $\chi_{984}(529,·)$, $\chi_{984}(25,·)$, $\chi_{984}(409,·)$, $\chi_{984}(197,·)$, $\chi_{984}(865,·)$, $\chi_{984}(869,·)$, $\chi_{984}(625,·)$, $\chi_{984}(389,·)$, $\chi_{984}(173,·)$, $\chi_{984}(893,·)$, $\chi_{984}(385,·)$, $\chi_{984}(433,·)$, $\chi_{984}(677,·)$, $\chi_{984}(769,·)$, $\chi_{984}(125,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{3888} a^{8} + \frac{1}{3}$, $\frac{1}{3888} a^{9} + \frac{1}{3} a$, $\frac{1}{23328} a^{10} + \frac{1}{18} a^{2}$, $\frac{1}{23328} a^{11} + \frac{1}{18} a^{3}$, $\frac{1}{139968} a^{12} + \frac{1}{108} a^{4}$, $\frac{1}{139968} a^{13} + \frac{1}{108} a^{5}$, $\frac{1}{839808} a^{14} + \frac{1}{648} a^{6}$, $\frac{1}{839808} a^{15} + \frac{1}{648} a^{7}$, $\frac{1}{1103507712} a^{16} + \frac{1}{10217664} a^{14} + \frac{35}{10217664} a^{12} - \frac{5}{851472} a^{10} + \frac{53}{851472} a^{8} - \frac{17}{3942} a^{4} - \frac{85}{1314} a^{2} - \frac{110}{657}$, $\frac{1}{1103507712} a^{17} + \frac{1}{10217664} a^{15} + \frac{35}{10217664} a^{13} - \frac{5}{851472} a^{11} + \frac{53}{851472} a^{9} - \frac{17}{3942} a^{5} - \frac{85}{1314} a^{3} - \frac{110}{657} a$, $\frac{1}{38884285798636032} a^{18} + \frac{1762651}{6480714299772672} a^{16} + \frac{46604785}{180019841660352} a^{14} - \frac{2294263}{5000551157232} a^{12} - \frac{1915439}{205502102352} a^{10} + \frac{21137090}{312534447327} a^{8} - \frac{281152655}{138904198812} a^{6} + \frac{75059731}{15433799868} a^{4} - \frac{1443614141}{23150699802} a^{2} + \frac{653515540}{3858449967}$, $\frac{1}{38884285798636032} a^{19} + \frac{1762651}{6480714299772672} a^{17} + \frac{46604785}{180019841660352} a^{15} - \frac{2294263}{5000551157232} a^{13} - \frac{1915439}{205502102352} a^{11} + \frac{21137090}{312534447327} a^{9} - \frac{281152655}{138904198812} a^{7} + \frac{75059731}{15433799868} a^{5} - \frac{1443614141}{23150699802} a^{3} + \frac{653515540}{3858449967} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{134644}$, which has order $17234432$ (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:  $9$
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:  \( 5104264.636551031 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

\(\Q(\sqrt{41}) \), 4.0.39698496.3, 5.5.2825761.1, 10.10.327381934393961.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41Data not computed