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Properties

Label 20.0.15441087535...5121.3
Degree $20$
Signature $[0, 10]$
Discriminant $11^{17}\cdot 1451^{3}$
Root discriminant $22.88$
Ramified primes $11, 1451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T432

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 33, -95, 218, -347, 527, -779, 1325, -2109, 3002, -3482, 3386, -2689, 1810, -996, 455, -160, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 44*x^18 - 160*x^17 + 455*x^16 - 996*x^15 + 1810*x^14 - 2689*x^13 + 3386*x^12 - 3482*x^11 + 3002*x^10 - 2109*x^9 + 1325*x^8 - 779*x^7 + 527*x^6 - 347*x^5 + 218*x^4 - 95*x^3 + 33*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 44*x^18 - 160*x^17 + 455*x^16 - 996*x^15 + 1810*x^14 - 2689*x^13 + 3386*x^12 - 3482*x^11 + 3002*x^10 - 2109*x^9 + 1325*x^8 - 779*x^7 + 527*x^6 - 347*x^5 + 218*x^4 - 95*x^3 + 33*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 44 x^{18} - 160 x^{17} + 455 x^{16} - 996 x^{15} + 1810 x^{14} - 2689 x^{13} + 3386 x^{12} - 3482 x^{11} + 3002 x^{10} - 2109 x^{9} + 1325 x^{8} - 779 x^{7} + 527 x^{6} - 347 x^{5} + 218 x^{4} - 95 x^{3} + 33 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1544108753590939768502655121=11^{17}\cdot 1451^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.88$
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:  $11, 1451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{622055139954320249} a^{19} + \frac{180835707859644164}{622055139954320249} a^{18} - \frac{5131513145887867}{14466398603588843} a^{17} + \frac{163949501008206699}{622055139954320249} a^{16} - \frac{102563270368656256}{622055139954320249} a^{15} - \frac{80785037561314756}{622055139954320249} a^{14} - \frac{154045530043373483}{622055139954320249} a^{13} - \frac{213323230362677910}{622055139954320249} a^{12} - \frac{195075828592925287}{622055139954320249} a^{11} + \frac{287403405831085680}{622055139954320249} a^{10} - \frac{231734622129893001}{622055139954320249} a^{9} + \frac{19023758332989228}{622055139954320249} a^{8} - \frac{27078381839957718}{622055139954320249} a^{7} + \frac{2124805478382814}{622055139954320249} a^{6} + \frac{246951147652132215}{622055139954320249} a^{5} + \frac{163803971977888905}{622055139954320249} a^{4} - \frac{85443639746895398}{622055139954320249} a^{3} + \frac{18213465018828500}{622055139954320249} a^{2} + \frac{171769336907480545}{622055139954320249} a + \frac{310088256250143651}{622055139954320249}$

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:  $9$
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:  \( 177270.754665 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T432:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n432 are not computed
Character table for t20n432 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.311034736331.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
1451Data not computed