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Properties

Label 20.0.15418170861...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3125, 0, 41250, 0, 80875, 0, 79750, 0, 49255, 0, 20592, 0, 6200, 0, 1386, 0, 221, 0, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 22*x^18 + 221*x^16 + 1386*x^14 + 6200*x^12 + 20592*x^10 + 49255*x^8 + 79750*x^6 + 80875*x^4 + 41250*x^2 + 3125)
 
gp: K = bnfinit(x^20 + 22*x^18 + 221*x^16 + 1386*x^14 + 6200*x^12 + 20592*x^10 + 49255*x^8 + 79750*x^6 + 80875*x^4 + 41250*x^2 + 3125, 1)
 

Normalized defining polynomial

\( x^{20} + 22 x^{18} + 221 x^{16} + 1386 x^{14} + 6200 x^{12} + 20592 x^{10} + 49255 x^{8} + 79750 x^{6} + 80875 x^{4} + 41250 x^{2} + 3125 \)

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:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{10} + \frac{1}{10} a^{8} + \frac{1}{10} a^{6} + \frac{1}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} + \frac{1}{10} a^{9} + \frac{1}{10} a^{7} + \frac{1}{5} a^{3} - \frac{1}{2} a$, $\frac{1}{50} a^{14} + \frac{1}{25} a^{12} + \frac{3}{25} a^{10} + \frac{8}{25} a^{8} - \frac{2}{5} a^{6} + \frac{17}{50} a^{4} - \frac{1}{5} a^{2} - \frac{1}{2}$, $\frac{1}{100} a^{15} - \frac{1}{100} a^{14} - \frac{3}{100} a^{13} - \frac{1}{50} a^{12} + \frac{21}{100} a^{11} - \frac{3}{50} a^{10} - \frac{7}{50} a^{9} - \frac{4}{25} a^{8} - \frac{1}{4} a^{7} - \frac{3}{10} a^{6} + \frac{21}{50} a^{5} + \frac{33}{100} a^{4} - \frac{9}{20} a^{3} - \frac{2}{5} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{500} a^{16} + \frac{1}{250} a^{14} - \frac{1}{20} a^{13} - \frac{19}{500} a^{12} + \frac{3}{20} a^{11} + \frac{29}{125} a^{10} - \frac{3}{10} a^{9} - \frac{39}{100} a^{8} + \frac{9}{20} a^{7} + \frac{21}{250} a^{6} - \frac{1}{4} a^{5} - \frac{21}{50} a^{4} + \frac{3}{20} a^{3} + \frac{7}{20} a^{2} + \frac{1}{4}$, $\frac{1}{500} a^{17} + \frac{1}{250} a^{15} - \frac{1}{100} a^{14} - \frac{19}{500} a^{13} + \frac{3}{100} a^{12} + \frac{29}{125} a^{11} + \frac{1}{25} a^{10} - \frac{39}{100} a^{9} + \frac{39}{100} a^{8} + \frac{21}{250} a^{7} - \frac{1}{4} a^{6} - \frac{21}{50} a^{5} + \frac{33}{100} a^{4} + \frac{7}{20} a^{3} - \frac{3}{10} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1691300597500} a^{18} - \frac{1557564373}{1691300597500} a^{16} - \frac{4890276347}{845650298750} a^{14} - \frac{1}{20} a^{13} - \frac{73386420609}{1691300597500} a^{12} - \frac{1}{10} a^{11} + \frac{30691912801}{338260119500} a^{10} + \frac{9}{20} a^{9} + \frac{717670095617}{1691300597500} a^{8} - \frac{1}{20} a^{7} + \frac{10246424179}{169130059750} a^{6} + \frac{7047124732}{16913005975} a^{4} - \frac{1}{10} a^{3} + \frac{163080829}{6765202390} a^{2} - \frac{1}{4} a + \frac{78037686}{676520239}$, $\frac{1}{1691300597500} a^{19} - \frac{1557564373}{1691300597500} a^{17} + \frac{7132453281}{1691300597500} a^{15} + \frac{11251155304}{422825149375} a^{13} + \frac{124251023}{169130059750} a^{11} - \frac{1}{4} a^{10} - \frac{195632227033}{1691300597500} a^{9} - \frac{1}{4} a^{8} - \frac{30246169567}{338260119500} a^{7} - \frac{1}{2} a^{6} + \frac{5694084254}{16913005975} a^{5} + \frac{1}{4} a^{4} + \frac{3708762853}{13530404780} a^{3} - \frac{1}{4} a^{2} - \frac{364369495}{2706080956} a - \frac{1}{4}$

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:  \( 1744254.52175 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.219503494144.2

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 R $20$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$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}$