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Properties

Label 20.0.89224910880...7504.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{2}\cdot 1871^{2}$
Root discriminant $39.58$
Ramified primes $2, 11, 23, 1871$
Class number $72$ (GRH)
Class group $[72]$ (GRH)
Galois group 20T341

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![279841, 0, 444360, 0, 289754, 0, 91406, 0, 18132, 0, 6464, 0, 3105, 0, 886, 0, 174, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 + 174*x^16 + 886*x^14 + 3105*x^12 + 6464*x^10 + 18132*x^8 + 91406*x^6 + 289754*x^4 + 444360*x^2 + 279841)
 
gp: K = bnfinit(x^20 + 20*x^18 + 174*x^16 + 886*x^14 + 3105*x^12 + 6464*x^10 + 18132*x^8 + 91406*x^6 + 289754*x^4 + 444360*x^2 + 279841, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} + 174 x^{16} + 886 x^{14} + 3105 x^{12} + 6464 x^{10} + 18132 x^{8} + 91406 x^{6} + 289754 x^{4} + 444360 x^{2} + 279841 \)

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:  \(89224910880483379681166705557504=2^{20}\cdot 11^{16}\cdot 23^{2}\cdot 1871^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.58$
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:  $2, 11, 23, 1871$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{4}{9} a^{6} + \frac{5}{18} a^{4} - \frac{1}{2} a^{2} + \frac{2}{9}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{4}{9} a^{7} + \frac{5}{18} a^{5} - \frac{1}{2} a^{3} + \frac{2}{9} a$, $\frac{1}{1242} a^{16} - \frac{1}{414} a^{14} + \frac{59}{1242} a^{12} + \frac{3}{46} a^{10} + \frac{7}{27} a^{8} - \frac{107}{414} a^{6} - \frac{199}{1242} a^{4} + \frac{209}{621} a^{2} + \frac{1}{27}$, $\frac{1}{1242} a^{17} - \frac{1}{414} a^{15} + \frac{59}{1242} a^{13} + \frac{3}{46} a^{11} + \frac{7}{27} a^{9} - \frac{107}{414} a^{7} - \frac{199}{1242} a^{5} + \frac{209}{621} a^{3} + \frac{1}{27} a$, $\frac{1}{24860410837470670806} a^{18} - \frac{4681742013887827}{12430205418735335403} a^{16} + \frac{428202188002872605}{24860410837470670806} a^{14} - \frac{1515647798023118623}{24860410837470670806} a^{12} + \frac{207485562220505843}{1080887427716116122} a^{10} + \frac{3215327676197526299}{12430205418735335403} a^{8} - \frac{2283533294310930823}{24860410837470670806} a^{6} - \frac{467869781850329380}{1381133935415037267} a^{4} + \frac{358104518177878915}{1080887427716116122} a^{2} - \frac{10982892270077507}{46995105552874614}$, $\frac{1}{571789449261825428538} a^{19} - \frac{64731043553672056}{285894724630912714269} a^{17} - \frac{15785109227738869225}{571789449261825428538} a^{15} + \frac{16258945457753013161}{571789449261825428538} a^{13} - \frac{738252860045417246}{12430205418735335403} a^{11} - \frac{20263949225858107240}{285894724630912714269} a^{9} + \frac{20205760050237827998}{285894724630912714269} a^{7} - \frac{3743892788076335651}{31766080514545857141} a^{5} - \frac{11912839709517159185}{24860410837470670806} a^{3} - \frac{53791971286604329}{540443713858058061} a$

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

Class group and class number

$C_{72}$, which has order $72$ (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:  \( 447363.730538 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T341:

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 t20n341 are not computed
Character table for t20n341 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.9445893863498752.2, 10.0.401065466351.1, 10.10.5048580365312.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1871Data not computed