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Properties

Label 20.0.99133055418...1808.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 7^{10}\cdot 17^{11}$
Root discriminant $17.78$
Ramified primes $2, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 13, 69, 176, 204, 83, 108, 165, 112, -55, 58, 73, -9, -18, 23, 10, -18, 7, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 4*x^18 + 7*x^17 - 18*x^16 + 10*x^15 + 23*x^14 - 18*x^13 - 9*x^12 + 73*x^11 + 58*x^10 - 55*x^9 + 112*x^8 + 165*x^7 + 108*x^6 + 83*x^5 + 204*x^4 + 176*x^3 + 69*x^2 + 13*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 4*x^18 + 7*x^17 - 18*x^16 + 10*x^15 + 23*x^14 - 18*x^13 - 9*x^12 + 73*x^11 + 58*x^10 - 55*x^9 + 112*x^8 + 165*x^7 + 108*x^6 + 83*x^5 + 204*x^4 + 176*x^3 + 69*x^2 + 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 4 x^{18} + 7 x^{17} - 18 x^{16} + 10 x^{15} + 23 x^{14} - 18 x^{13} - 9 x^{12} + 73 x^{11} + 58 x^{10} - 55 x^{9} + 112 x^{8} + 165 x^{7} + 108 x^{6} + 83 x^{5} + 204 x^{4} + 176 x^{3} + 69 x^{2} + 13 x + 1 \)

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:  \(9913305541837631770231808=2^{10}\cdot 7^{10}\cdot 17^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.78$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{15} + \frac{1}{7} a^{14} - \frac{2}{7} a^{12} + \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{6778962099106967} a^{19} + \frac{258814695340720}{6778962099106967} a^{18} - \frac{56192938428279}{968423157015281} a^{17} - \frac{455837169265760}{6778962099106967} a^{16} - \frac{3188850766365434}{6778962099106967} a^{15} + \frac{1301037345058816}{6778962099106967} a^{14} + \frac{2116385229660143}{6778962099106967} a^{13} - \frac{1551550065588232}{6778962099106967} a^{12} - \frac{2765389660404603}{6778962099106967} a^{11} + \frac{126304707694650}{6778962099106967} a^{10} - \frac{11142416323142}{968423157015281} a^{9} + \frac{961620969177560}{6778962099106967} a^{8} + \frac{330860214634434}{968423157015281} a^{7} - \frac{3327349718601471}{6778962099106967} a^{6} + \frac{870817337471727}{6778962099106967} a^{5} + \frac{1923695204865345}{6778962099106967} a^{4} - \frac{164848681907176}{616269281736997} a^{3} - \frac{16762241176124}{616269281736997} a^{2} + \frac{3166050524713607}{6778962099106967} a - \frac{638383149978477}{6778962099106967}$

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:  \( 12218.987562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 5.1.14161.1, 10.2.3409076657.1

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 R $20$ $20$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$