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Properties

Label 20.0.22950236375...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{12}\cdot 13^{6}\cdot 41^{7}$
Root discriminant $20.80$
Ramified primes $5, 13, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![739, -3165, 8748, -10196, 12853, -12550, 10907, -7446, 4352, -2745, 1710, -1265, 803, -432, 187, -90, 62, -37, 18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 18*x^18 - 37*x^17 + 62*x^16 - 90*x^15 + 187*x^14 - 432*x^13 + 803*x^12 - 1265*x^11 + 1710*x^10 - 2745*x^9 + 4352*x^8 - 7446*x^7 + 10907*x^6 - 12550*x^5 + 12853*x^4 - 10196*x^3 + 8748*x^2 - 3165*x + 739)
 
gp: K = bnfinit(x^20 - 5*x^19 + 18*x^18 - 37*x^17 + 62*x^16 - 90*x^15 + 187*x^14 - 432*x^13 + 803*x^12 - 1265*x^11 + 1710*x^10 - 2745*x^9 + 4352*x^8 - 7446*x^7 + 10907*x^6 - 12550*x^5 + 12853*x^4 - 10196*x^3 + 8748*x^2 - 3165*x + 739, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 18 x^{18} - 37 x^{17} + 62 x^{16} - 90 x^{15} + 187 x^{14} - 432 x^{13} + 803 x^{12} - 1265 x^{11} + 1710 x^{10} - 2745 x^{9} + 4352 x^{8} - 7446 x^{7} + 10907 x^{6} - 12550 x^{5} + 12853 x^{4} - 10196 x^{3} + 8748 x^{2} - 3165 x + 739 \)

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:  \(229502363759100519775390625=5^{12}\cdot 13^{6}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.80$
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:  $5, 13, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{41} a^{18} + \frac{11}{41} a^{17} + \frac{13}{41} a^{16} - \frac{16}{41} a^{15} - \frac{5}{41} a^{14} + \frac{20}{41} a^{13} + \frac{18}{41} a^{12} + \frac{8}{41} a^{11} + \frac{10}{41} a^{10} - \frac{11}{41} a^{9} + \frac{11}{41} a^{8} - \frac{4}{41} a^{7} + \frac{1}{41} a^{6} + \frac{18}{41} a^{5} - \frac{15}{41} a^{4} - \frac{17}{41} a^{3} + \frac{3}{41} a^{2} - \frac{19}{41} a - \frac{12}{41}$, $\frac{1}{2787522104772971577049309045676274781} a^{19} + \frac{11820321015180833409729395653784312}{2787522104772971577049309045676274781} a^{18} + \frac{1096171970816267882668724170687909968}{2787522104772971577049309045676274781} a^{17} - \frac{720522382750409086261889341331565987}{2787522104772971577049309045676274781} a^{16} - \frac{1346613918484808309926120492837538802}{2787522104772971577049309045676274781} a^{15} - \frac{1126592760573221584879039679459431473}{2787522104772971577049309045676274781} a^{14} - \frac{179650114410605468091765191809969332}{2787522104772971577049309045676274781} a^{13} + \frac{1126772221934520873119418409415191341}{2787522104772971577049309045676274781} a^{12} + \frac{1371424296308471976418476511406579909}{2787522104772971577049309045676274781} a^{11} + \frac{1193991311337066372587894888237034190}{2787522104772971577049309045676274781} a^{10} + \frac{956041810298196891019685092926105636}{2787522104772971577049309045676274781} a^{9} + \frac{707873542584528270718223387147134507}{2787522104772971577049309045676274781} a^{8} + \frac{373626541174101510872073484098517825}{2787522104772971577049309045676274781} a^{7} + \frac{661706702620404092064709195872327894}{2787522104772971577049309045676274781} a^{6} + \frac{380465379825399788083052540722871867}{2787522104772971577049309045676274781} a^{5} + \frac{1054582510354020141939831445043845022}{2787522104772971577049309045676274781} a^{4} - \frac{945306134900314045401933200169435940}{2787522104772971577049309045676274781} a^{3} + \frac{817918220144398092513923869178144598}{2787522104772971577049309045676274781} a^{2} + \frac{40489212918999002324063470904678927}{2787522104772971577049309045676274781} a + \frac{1191074209931240319728627974374949924}{2787522104772971577049309045676274781}$

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:  $9$
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:  \( 62009.3204021 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.2665.1, 10.2.887778125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.12.0.1$x^{12} + x^{2} - x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
41Data not computed