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Properties

Label 20.20.2184548182...3969.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 109^{2}\cdot 7369^{3}$
Root discriminant $41.40$
Ramified primes $11, 109, 7369$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T335

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 224, 208, -2028, -5804, 189, 20826, 26600, -4375, -29655, -11678, 12524, 8718, -2356, -2653, 150, 411, 8, -32, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 32*x^18 + 8*x^17 + 411*x^16 + 150*x^15 - 2653*x^14 - 2356*x^13 + 8718*x^12 + 12524*x^11 - 11678*x^10 - 29655*x^9 - 4375*x^8 + 26600*x^7 + 20826*x^6 + 189*x^5 - 5804*x^4 - 2028*x^3 + 208*x^2 + 224*x + 32)
 
gp: K = bnfinit(x^20 - x^19 - 32*x^18 + 8*x^17 + 411*x^16 + 150*x^15 - 2653*x^14 - 2356*x^13 + 8718*x^12 + 12524*x^11 - 11678*x^10 - 29655*x^9 - 4375*x^8 + 26600*x^7 + 20826*x^6 + 189*x^5 - 5804*x^4 - 2028*x^3 + 208*x^2 + 224*x + 32, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 32 x^{18} + 8 x^{17} + 411 x^{16} + 150 x^{15} - 2653 x^{14} - 2356 x^{13} + 8718 x^{12} + 12524 x^{11} - 11678 x^{10} - 29655 x^{9} - 4375 x^{8} + 26600 x^{7} + 20826 x^{6} + 189 x^{5} - 5804 x^{4} - 2028 x^{3} + 208 x^{2} + 224 x + 32 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(218454818255016653811965258323969=11^{16}\cdot 109^{2}\cdot 7369^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.40$
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:  $11, 109, 7369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{90650192} a^{19} + \frac{4736555}{90650192} a^{18} + \frac{623382}{5665637} a^{17} - \frac{4852829}{22662548} a^{16} - \frac{12179557}{90650192} a^{15} - \frac{3074043}{45325096} a^{14} + \frac{8656735}{90650192} a^{13} - \frac{1421077}{11331274} a^{12} + \frac{4360853}{45325096} a^{11} + \frac{3837349}{22662548} a^{10} - \frac{19302219}{45325096} a^{9} + \frac{6196377}{90650192} a^{8} + \frac{36590165}{90650192} a^{7} - \frac{57621}{131759} a^{6} - \frac{20853789}{45325096} a^{5} - \frac{32053451}{90650192} a^{4} + \frac{1941295}{11331274} a^{3} - \frac{1587839}{11331274} a^{2} + \frac{1152651}{11331274} a - \frac{2217437}{5665637}$

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:  $19$
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:  \( 5038822745.78 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T335:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 224 conjugacy class representatives for t20n335 are not computed
Character table for t20n335 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1579610594089.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.2.2$x^{4} - 109 x^{2} + 71286$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7369Data not computed