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Properties

Label 20.12.1077966233...2544.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 11^{18}\cdot 43^{2}$
Root discriminant $25.21$
Ramified primes $2, 11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -35, 190, -97, -1118, 2973, -2636, -1632, 6786, -7888, 4338, 200, -2654, 2704, -1676, 745, -246, 59, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 59*x^18 - 246*x^17 + 745*x^16 - 1676*x^15 + 2704*x^14 - 2654*x^13 + 200*x^12 + 4338*x^11 - 7888*x^10 + 6786*x^9 - 1632*x^8 - 2636*x^7 + 2973*x^6 - 1118*x^5 - 97*x^4 + 190*x^3 - 35*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 59*x^18 - 246*x^17 + 745*x^16 - 1676*x^15 + 2704*x^14 - 2654*x^13 + 200*x^12 + 4338*x^11 - 7888*x^10 + 6786*x^9 - 1632*x^8 - 2636*x^7 + 2973*x^6 - 1118*x^5 - 97*x^4 + 190*x^3 - 35*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 59 x^{18} - 246 x^{17} + 745 x^{16} - 1676 x^{15} + 2704 x^{14} - 2654 x^{13} + 200 x^{12} + 4338 x^{11} - 7888 x^{10} + 6786 x^{9} - 1632 x^{8} - 2636 x^{7} + 2973 x^{6} - 1118 x^{5} - 97 x^{4} + 190 x^{3} - 35 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10779662339431083287111532544=2^{20}\cdot 11^{18}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.21$
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:  $2, 11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{43} a^{16} - \frac{8}{43} a^{15} + \frac{10}{43} a^{14} - \frac{16}{43} a^{13} - \frac{18}{43} a^{12} - \frac{5}{43} a^{11} - \frac{7}{43} a^{10} + \frac{15}{43} a^{9} - \frac{13}{43} a^{8} + \frac{18}{43} a^{7} + \frac{6}{43} a^{6} + \frac{1}{43} a^{5} - \frac{12}{43} a^{4} + \frac{6}{43} a^{3} + \frac{6}{43} a^{2} + \frac{16}{43} a - \frac{8}{43}$, $\frac{1}{43} a^{17} - \frac{11}{43} a^{15} + \frac{21}{43} a^{14} - \frac{17}{43} a^{13} - \frac{20}{43} a^{12} - \frac{4}{43} a^{11} + \frac{2}{43} a^{10} + \frac{21}{43} a^{9} + \frac{21}{43} a^{7} + \frac{6}{43} a^{6} - \frac{4}{43} a^{5} - \frac{4}{43} a^{4} + \frac{11}{43} a^{3} + \frac{21}{43} a^{2} - \frac{9}{43} a - \frac{21}{43}$, $\frac{1}{914072543} a^{18} - \frac{9}{914072543} a^{17} - \frac{4387667}{914072543} a^{16} + \frac{35101540}{914072543} a^{15} + \frac{254240385}{914072543} a^{14} + \frac{348257270}{914072543} a^{13} - \frac{124523859}{914072543} a^{12} - \frac{354933548}{914072543} a^{11} - \frac{157876149}{914072543} a^{10} + \frac{238167021}{914072543} a^{9} - \frac{425980956}{914072543} a^{8} + \frac{248274369}{914072543} a^{7} + \frac{328804341}{914072543} a^{6} - \frac{142385589}{914072543} a^{5} + \frac{52575723}{914072543} a^{4} - \frac{177607109}{914072543} a^{3} + \frac{83842962}{914072543} a^{2} - \frac{201568726}{914072543} a - \frac{140040659}{914072543}$, $\frac{1}{914072543} a^{19} - \frac{4387748}{914072543} a^{17} - \frac{4387463}{914072543} a^{16} - \frac{343918298}{914072543} a^{15} - \frac{105796894}{914072543} a^{14} + \frac{267573942}{914072543} a^{13} + \frac{352496807}{914072543} a^{12} + \frac{304012091}{914072543} a^{11} - \frac{268645777}{914072543} a^{10} - \frac{110622853}{914072543} a^{9} + \frac{70735937}{914072543} a^{8} - \frac{178943967}{914072543} a^{7} + \frac{74635851}{914072543} a^{6} - \frac{314822035}{914072543} a^{5} + \frac{295574398}{914072543} a^{4} + \frac{313524067}{914072543} a^{3} - \frac{361054611}{914072543} a^{2} - \frac{126014107}{914072543} a - \frac{346293388}{914072543}$

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:  $15$
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:  \( 5285142.41719 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
11Data not computed
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$