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Properties

Label 20.4.21800000000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 5^{23}\cdot 109$
Root discriminant $18.49$
Ramified primes $2, 5, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 90, -85, -210, 425, 82, -680, 190, 675, -440, -301, 240, 195, -160, -50, 38, 25, -10, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 10*x^17 + 25*x^16 + 38*x^15 - 50*x^14 - 160*x^13 + 195*x^12 + 240*x^11 - 301*x^10 - 440*x^9 + 675*x^8 + 190*x^7 - 680*x^6 + 82*x^5 + 425*x^4 - 210*x^3 - 85*x^2 + 90*x - 19)
 
gp: K = bnfinit(x^20 - 5*x^18 - 10*x^17 + 25*x^16 + 38*x^15 - 50*x^14 - 160*x^13 + 195*x^12 + 240*x^11 - 301*x^10 - 440*x^9 + 675*x^8 + 190*x^7 - 680*x^6 + 82*x^5 + 425*x^4 - 210*x^3 - 85*x^2 + 90*x - 19, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 10 x^{17} + 25 x^{16} + 38 x^{15} - 50 x^{14} - 160 x^{13} + 195 x^{12} + 240 x^{11} - 301 x^{10} - 440 x^{9} + 675 x^{8} + 190 x^{7} - 680 x^{6} + 82 x^{5} + 425 x^{4} - 210 x^{3} - 85 x^{2} + 90 x - 19 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21800000000000000000000000=2^{24}\cdot 5^{23}\cdot 109\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.49$
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:  $2, 5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{10}{23} a^{17} + \frac{11}{23} a^{16} - \frac{4}{23} a^{15} + \frac{4}{23} a^{14} + \frac{5}{23} a^{12} + \frac{5}{23} a^{11} + \frac{9}{23} a^{10} + \frac{2}{23} a^{9} - \frac{2}{23} a^{8} - \frac{7}{23} a^{7} - \frac{9}{23} a^{6} - \frac{2}{23} a^{5} + \frac{10}{23} a^{4} + \frac{5}{23} a^{3} + \frac{3}{23} a^{2} - \frac{2}{23} a + \frac{11}{23}$, $\frac{1}{1125032567857127} a^{19} + \frac{7056004525901}{1125032567857127} a^{18} + \frac{293336386579444}{1125032567857127} a^{17} + \frac{201711218833470}{1125032567857127} a^{16} - \frac{181427392059778}{1125032567857127} a^{15} - \frac{245615781627714}{1125032567857127} a^{14} + \frac{7063860638195}{59212240413533} a^{13} - \frac{152372527887830}{1125032567857127} a^{12} - \frac{13864801242488}{48914459472049} a^{11} - \frac{417249643104771}{1125032567857127} a^{10} - \frac{420556743487562}{1125032567857127} a^{9} + \frac{398099317789197}{1125032567857127} a^{8} + \frac{158606774567520}{1125032567857127} a^{7} - \frac{208924400706293}{1125032567857127} a^{6} - \frac{351855704860225}{1125032567857127} a^{5} + \frac{156209824644151}{1125032567857127} a^{4} - \frac{543251226839043}{1125032567857127} a^{3} + \frac{192625424682660}{1125032567857127} a^{2} + \frac{428692502733397}{1125032567857127} a - \frac{24807691896629}{59212240413533}$

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:  $11$
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:  \( 83224.2834349 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.50000.1, 10.2.12500000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$