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Properties

Label 20.16.1742309603...0625.1
Degree $20$
Signature $[16, 2]$
Discriminant $5^{10}\cdot 11^{2}\cdot 135049\cdot 10448989^{2}$
Root discriminant $25.83$
Ramified primes $5, 11, 135049, 10448989$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1045

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, -20, 120, 98, -562, -279, 1359, 453, -2071, -225, 2011, -307, -1121, 461, 276, -213, 6, 31, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 31*x^18 + 6*x^17 - 213*x^16 + 276*x^15 + 461*x^14 - 1121*x^13 - 307*x^12 + 2011*x^11 - 225*x^10 - 2071*x^9 + 453*x^8 + 1359*x^7 - 279*x^6 - 562*x^5 + 98*x^4 + 120*x^3 - 20*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 31*x^18 + 6*x^17 - 213*x^16 + 276*x^15 + 461*x^14 - 1121*x^13 - 307*x^12 + 2011*x^11 - 225*x^10 - 2071*x^9 + 453*x^8 + 1359*x^7 - 279*x^6 - 562*x^5 + 98*x^4 + 120*x^3 - 20*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 31 x^{18} + 6 x^{17} - 213 x^{16} + 276 x^{15} + 461 x^{14} - 1121 x^{13} - 307 x^{12} + 2011 x^{11} - 225 x^{10} - 2071 x^{9} + 453 x^{8} + 1359 x^{7} - 279 x^{6} - 562 x^{5} + 98 x^{4} + 120 x^{3} - 20 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17423096031535445218837890625=5^{10}\cdot 11^{2}\cdot 135049\cdot 10448989^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.83$
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:  $5, 11, 135049, 10448989$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{8}{17} a^{17} + \frac{2}{17} a^{16} + \frac{1}{17} a^{15} + \frac{3}{17} a^{14} + \frac{4}{17} a^{13} - \frac{3}{17} a^{12} + \frac{3}{17} a^{11} - \frac{6}{17} a^{10} + \frac{7}{17} a^{9} + \frac{4}{17} a^{8} + \frac{3}{17} a^{7} + \frac{2}{17} a^{6} - \frac{8}{17} a^{5} - \frac{4}{17} a^{4} + \frac{2}{17} a^{3} - \frac{7}{17} a^{2} + \frac{5}{17} a + \frac{6}{17}$, $\frac{1}{17} a^{19} + \frac{6}{17} a^{17} + \frac{2}{17} a^{16} - \frac{5}{17} a^{15} - \frac{3}{17} a^{14} - \frac{1}{17} a^{13} - \frac{7}{17} a^{12} + \frac{4}{17} a^{11} + \frac{4}{17} a^{10} - \frac{1}{17} a^{9} + \frac{5}{17} a^{8} - \frac{5}{17} a^{7} - \frac{7}{17} a^{6} - \frac{8}{17} a^{5} - \frac{6}{17} a^{3} - \frac{7}{17} a^{2} + \frac{3}{17}$

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:  $17$
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:  \( 16265111.0838 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1045:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 29491200
The 702 conjugacy class representatives for t20n1045 are not computed
Character table for t20n1045 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.8.359183996875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R $20$ R $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
135049Data not computed
10448989Data not computed