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Properties

Label 20.4.34884167202...8921.1
Degree $20$
Signature $[4, 8]$
Discriminant $11^{18}\cdot 89^{4}$
Root discriminant $21.24$
Ramified primes $11, 89$
Class number $1$
Class group Trivial
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 17, -34, 1, 69, -45, -107, 4, 110, -10, -44, -96, -48, 40, 19, 3, -2, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^18 - 2*x^17 + 3*x^16 + 19*x^15 + 40*x^14 - 48*x^13 - 96*x^12 - 44*x^11 - 10*x^10 + 110*x^9 + 4*x^8 - 107*x^7 - 45*x^6 + 69*x^5 + x^4 - 34*x^3 + 17*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 3*x^18 - 2*x^17 + 3*x^16 + 19*x^15 + 40*x^14 - 48*x^13 - 96*x^12 - 44*x^11 - 10*x^10 + 110*x^9 + 4*x^8 - 107*x^7 - 45*x^6 + 69*x^5 + x^4 - 34*x^3 + 17*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 3 x^{18} - 2 x^{17} + 3 x^{16} + 19 x^{15} + 40 x^{14} - 48 x^{13} - 96 x^{12} - 44 x^{11} - 10 x^{10} + 110 x^{9} + 4 x^{8} - 107 x^{7} - 45 x^{6} + 69 x^{5} + x^{4} - 34 x^{3} + 17 x^{2} - 5 x + 1 \)

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:  \(348841672023202139208688921=11^{18}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.24$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1438974044508037316533} a^{19} - \frac{311776669417527204555}{1438974044508037316533} a^{18} + \frac{569124840817380105005}{1438974044508037316533} a^{17} + \frac{455613613341019762788}{1438974044508037316533} a^{16} + \frac{183911165847798529661}{1438974044508037316533} a^{15} + \frac{501268416751311542857}{1438974044508037316533} a^{14} + \frac{704909702687891021634}{1438974044508037316533} a^{13} + \frac{27103833216469942481}{1438974044508037316533} a^{12} - \frac{362061128433270989389}{1438974044508037316533} a^{11} - \frac{281558326399780078529}{1438974044508037316533} a^{10} + \frac{51901764465343922677}{1438974044508037316533} a^{9} - \frac{630314561682078575191}{1438974044508037316533} a^{8} - \frac{23660965039549211804}{62564088891653796371} a^{7} - \frac{404680468623440450632}{1438974044508037316533} a^{6} - \frac{251716550121300437195}{1438974044508037316533} a^{5} - \frac{70055911442445667456}{1438974044508037316533} a^{4} + \frac{240622642718117276271}{1438974044508037316533} a^{3} - \frac{525961524115781479239}{1438974044508037316533} a^{2} + \frac{537766675194193613254}{1438974044508037316533} a + \frac{391144541370918268325}{1438974044508037316533}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 195254.149815 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1, 10.4.18677303660411.1, 10.8.209857344499.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed