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Properties

Label 20.8.12396113234...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{13}\cdot 6329^{5}$
Root discriminant $25.39$
Ramified primes $5, 6329$
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![116, -322, 1135, -1522, -2179, 4388, -1350, -3046, 2655, -94, -773, 529, -46, -69, 8, 11, -8, -5, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 4*x^18 - 5*x^17 - 8*x^16 + 11*x^15 + 8*x^14 - 69*x^13 - 46*x^12 + 529*x^11 - 773*x^10 - 94*x^9 + 2655*x^8 - 3046*x^7 - 1350*x^6 + 4388*x^5 - 2179*x^4 - 1522*x^3 + 1135*x^2 - 322*x + 116)
 
gp: K = bnfinit(x^20 - 2*x^19 + 4*x^18 - 5*x^17 - 8*x^16 + 11*x^15 + 8*x^14 - 69*x^13 - 46*x^12 + 529*x^11 - 773*x^10 - 94*x^9 + 2655*x^8 - 3046*x^7 - 1350*x^6 + 4388*x^5 - 2179*x^4 - 1522*x^3 + 1135*x^2 - 322*x + 116, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 4 x^{18} - 5 x^{17} - 8 x^{16} + 11 x^{15} + 8 x^{14} - 69 x^{13} - 46 x^{12} + 529 x^{11} - 773 x^{10} - 94 x^{9} + 2655 x^{8} - 3046 x^{7} - 1350 x^{6} + 4388 x^{5} - 2179 x^{4} - 1522 x^{3} + 1135 x^{2} - 322 x + 116 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12396113234941360655517578125=5^{13}\cdot 6329^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.39$
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:  $5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{21} a^{18} - \frac{1}{7} a^{17} + \frac{1}{21} a^{16} + \frac{5}{21} a^{15} - \frac{5}{21} a^{14} - \frac{1}{3} a^{13} - \frac{4}{21} a^{12} + \frac{5}{21} a^{11} + \frac{8}{21} a^{10} + \frac{8}{21} a^{9} - \frac{1}{7} a^{8} + \frac{1}{21} a^{7} + \frac{5}{21} a^{6} + \frac{3}{7} a^{5} - \frac{10}{21} a^{4} - \frac{1}{7} a^{3} + \frac{5}{21} a^{2} + \frac{10}{21} a + \frac{10}{21}$, $\frac{1}{158668504884976215606360010638} a^{19} - \frac{406829206838702455031977616}{79334252442488107803180005319} a^{18} + \frac{12435311820393264046510205441}{79334252442488107803180005319} a^{17} + \frac{47837422586178038509749258067}{158668504884976215606360010638} a^{16} - \frac{35726871187247404683550587839}{79334252442488107803180005319} a^{15} - \frac{1953601036948764339959921111}{158668504884976215606360010638} a^{14} - \frac{1491276785795011204352823386}{26444750814162702601060001773} a^{13} - \frac{12406541360353450401723849069}{52889501628325405202120003546} a^{12} - \frac{53930811167151254106235420}{3777821544880386085865714539} a^{11} - \frac{25115727661355728064785778279}{52889501628325405202120003546} a^{10} - \frac{20529929885214574945798874405}{158668504884976215606360010638} a^{9} - \frac{10191282086392770489442279189}{79334252442488107803180005319} a^{8} + \frac{7706724188075719193412514519}{158668504884976215606360010638} a^{7} - \frac{20162247461437571034944109217}{79334252442488107803180005319} a^{6} - \frac{328892522168143713578863996}{702073030464496529231681463} a^{5} - \frac{34988177109699929038753238917}{79334252442488107803180005319} a^{4} + \frac{7957411397255480163722569943}{22666929269282316515194287234} a^{3} + \frac{31170091937937671225556241438}{79334252442488107803180005319} a^{2} - \frac{13898140289124925084693123047}{52889501628325405202120003546} a + \frac{273258142628490656862201325}{79334252442488107803180005319}$

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:  $13$
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:  \( 5575342.11676 \) (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

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 $16{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
6329Data not computed