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Properties

Label 20.2.10658901185...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{4}\cdot 5^{20}\cdot 7^{8}\cdot 59^{4}$
Root discriminant $28.27$
Ramified primes $2, 5, 7, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -50, 165, -85, 125, -187, -5, -195, 100, -65, 63, -70, -25, -50, -35, 28, 30, -10, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 10*x^17 + 30*x^16 + 28*x^15 - 35*x^14 - 50*x^13 - 25*x^12 - 70*x^11 + 63*x^10 - 65*x^9 + 100*x^8 - 195*x^7 - 5*x^6 - 187*x^5 + 125*x^4 - 85*x^3 + 165*x^2 - 50*x + 31)
 
gp: K = bnfinit(x^20 - 10*x^18 - 10*x^17 + 30*x^16 + 28*x^15 - 35*x^14 - 50*x^13 - 25*x^12 - 70*x^11 + 63*x^10 - 65*x^9 + 100*x^8 - 195*x^7 - 5*x^6 - 187*x^5 + 125*x^4 - 85*x^3 + 165*x^2 - 50*x + 31, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 10 x^{17} + 30 x^{16} + 28 x^{15} - 35 x^{14} - 50 x^{13} - 25 x^{12} - 70 x^{11} + 63 x^{10} - 65 x^{9} + 100 x^{8} - 195 x^{7} - 5 x^{6} - 187 x^{5} + 125 x^{4} - 85 x^{3} + 165 x^{2} - 50 x + 31 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-106589011856324768066406250000=-\,2^{4}\cdot 5^{20}\cdot 7^{8}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.27$
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:  $2, 5, 7, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} - \frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{2}{7} a^{13} - \frac{2}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{4} + \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{14} a^{18} - \frac{1}{14} a^{16} - \frac{1}{7} a^{15} + \frac{5}{14} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{2} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{5}{14} a^{7} - \frac{1}{7} a^{5} - \frac{1}{2} a^{4} - \frac{5}{14} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{1}{2}$, $\frac{1}{136231283536125753447458} a^{19} + \frac{988198914630823299779}{68115641768062876723729} a^{18} - \frac{5950670456701666936623}{136231283536125753447458} a^{17} - \frac{422411294818054899800}{9730805966866125246247} a^{16} - \frac{10846776769365007686013}{136231283536125753447458} a^{15} - \frac{32049453681259752081138}{68115641768062876723729} a^{14} + \frac{18869269655368571481661}{68115641768062876723729} a^{13} - \frac{17609948007449911992806}{68115641768062876723729} a^{12} - \frac{7754735886021914473585}{19461611933732250492494} a^{11} - \frac{20562791160100341432388}{68115641768062876723729} a^{10} + \frac{20641488991875052856083}{68115641768062876723729} a^{9} - \frac{20963378056017093846313}{136231283536125753447458} a^{8} - \frac{12181506497678029787294}{68115641768062876723729} a^{7} - \frac{22029389879675875138810}{68115641768062876723729} a^{6} + \frac{9726225904407016092129}{136231283536125753447458} a^{5} - \frac{4049631486824914718513}{136231283536125753447458} a^{4} - \frac{873787115894164207940}{68115641768062876723729} a^{3} + \frac{18097308007442665832538}{68115641768062876723729} a^{2} - \frac{5608096005971878846839}{136231283536125753447458} a - \frac{20141830995724900959989}{68115641768062876723729}$

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:  $10$
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:  \( 3262043.43252 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.1665712890625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
5Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$59$59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.6.4.1$x^{6} + 295 x^{3} + 27848$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
59.10.0.1$x^{10} + x^{2} - x + 37$$1$$10$$0$$C_{10}$$[\ ]^{10}$