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Properties

Label 20.8.65861868411...0144.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -25, -101, -173, -136, 53, 284, 425, 60, -172, -130, -222, 214, 99, -199, 106, 8, -48, 31, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 31*x^18 - 48*x^17 + 8*x^16 + 106*x^15 - 199*x^14 + 99*x^13 + 214*x^12 - 222*x^11 - 130*x^10 - 172*x^9 + 60*x^8 + 425*x^7 + 284*x^6 + 53*x^5 - 136*x^4 - 173*x^3 - 101*x^2 - 25*x + 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 31*x^18 - 48*x^17 + 8*x^16 + 106*x^15 - 199*x^14 + 99*x^13 + 214*x^12 - 222*x^11 - 130*x^10 - 172*x^9 + 60*x^8 + 425*x^7 + 284*x^6 + 53*x^5 - 136*x^4 - 173*x^3 - 101*x^2 - 25*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 31 x^{18} - 48 x^{17} + 8 x^{16} + 106 x^{15} - 199 x^{14} + 99 x^{13} + 214 x^{12} - 222 x^{11} - 130 x^{10} - 172 x^{9} + 60 x^{8} + 425 x^{7} + 284 x^{6} + 53 x^{5} - 136 x^{4} - 173 x^{3} - 101 x^{2} - 25 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(658618684118710741619590144=2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{7743213202903193926} a^{19} - \frac{664170234457224981}{7743213202903193926} a^{18} - \frac{903469055283414815}{7743213202903193926} a^{17} - \frac{143902580956906595}{7743213202903193926} a^{16} + \frac{1051942473706116203}{7743213202903193926} a^{15} + \frac{331460861502461285}{7743213202903193926} a^{14} - \frac{1014150370474344183}{7743213202903193926} a^{13} + \frac{1185456951791959343}{7743213202903193926} a^{12} + \frac{831710347866696305}{7743213202903193926} a^{11} + \frac{1751956536434612698}{3871606601451596963} a^{10} - \frac{2783154872958471247}{7743213202903193926} a^{9} - \frac{624057237881663355}{7743213202903193926} a^{8} + \frac{151057938046187378}{3871606601451596963} a^{7} - \frac{1189453734545006509}{7743213202903193926} a^{6} + \frac{1922180502295159997}{3871606601451596963} a^{5} - \frac{760184805723709996}{3871606601451596963} a^{4} + \frac{607433134775188499}{3871606601451596963} a^{3} - \frac{622824130490316075}{3871606601451596963} a^{2} + \frac{2333780696475243415}{7743213202903193926} a - \frac{1933087350002027974}{3871606601451596963}$

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:  \( 606854.159255 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
241Data not computed