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Properties

Label 20.10.2790564977...1875.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 11^{5}\cdot 36497^{4}$
Root discriminant $33.29$
Ramified primes $5, 11, 36497$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 20, -31, -69, 162, 110, -384, -557, 929, 817, -1110, -88, 383, -61, -81, -5, 27, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^18 + 27*x^17 - 5*x^16 - 81*x^15 - 61*x^14 + 383*x^13 - 88*x^12 - 1110*x^11 + 817*x^10 + 929*x^9 - 557*x^8 - 384*x^7 + 110*x^6 + 162*x^5 - 69*x^4 - 31*x^3 + 20*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 3*x^18 + 27*x^17 - 5*x^16 - 81*x^15 - 61*x^14 + 383*x^13 - 88*x^12 - 1110*x^11 + 817*x^10 + 929*x^9 - 557*x^8 - 384*x^7 + 110*x^6 + 162*x^5 - 69*x^4 - 31*x^3 + 20*x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 3 x^{18} + 27 x^{17} - 5 x^{16} - 81 x^{15} - 61 x^{14} + 383 x^{13} - 88 x^{12} - 1110 x^{11} + 817 x^{10} + 929 x^{9} - 557 x^{8} - 384 x^{7} + 110 x^{6} + 162 x^{5} - 69 x^{4} - 31 x^{3} + 20 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2790564977304713540069638671875=-\,5^{10}\cdot 11^{5}\cdot 36497^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.29$
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:  $5, 11, 36497$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{155} a^{18} + \frac{12}{155} a^{17} + \frac{16}{155} a^{16} + \frac{13}{155} a^{15} - \frac{37}{155} a^{14} - \frac{56}{155} a^{13} + \frac{18}{155} a^{12} - \frac{11}{155} a^{11} + \frac{21}{155} a^{10} + \frac{48}{155} a^{9} + \frac{13}{31} a^{8} - \frac{43}{155} a^{7} + \frac{6}{155} a^{6} + \frac{2}{5} a^{5} - \frac{71}{155} a^{4} + \frac{1}{31} a^{3} + \frac{8}{31} a^{2} - \frac{61}{155} a + \frac{26}{155}$, $\frac{1}{736346197519170182997965} a^{19} + \frac{301843896441176333548}{736346197519170182997965} a^{18} - \frac{177882082644099562544987}{736346197519170182997965} a^{17} + \frac{53396965889685682264314}{736346197519170182997965} a^{16} + \frac{64578414017079943581441}{736346197519170182997965} a^{15} + \frac{65748408303475917316272}{736346197519170182997965} a^{14} - \frac{11795321766769542531808}{66940563410833652999815} a^{13} - \frac{104910575991243498686608}{736346197519170182997965} a^{12} + \frac{15810049698733306482854}{147269239503834036599593} a^{11} - \frac{358700481066979457591346}{736346197519170182997965} a^{10} + \frac{296089942697346282117628}{736346197519170182997965} a^{9} - \frac{61213319092041853403003}{736346197519170182997965} a^{8} - \frac{89440634650680015162607}{736346197519170182997965} a^{7} - \frac{324591405092204077355332}{736346197519170182997965} a^{6} + \frac{364159388378302718620456}{736346197519170182997965} a^{5} + \frac{6465011186329032653464}{66940563410833652999815} a^{4} - \frac{20347873111883166471891}{147269239503834036599593} a^{3} - \frac{193545249480686979086291}{736346197519170182997965} a^{2} + \frac{35366419286900218508313}{147269239503834036599593} a - \frac{12084252744369649424994}{736346197519170182997965}$

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:  $14$
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:  \( 73820905.3723 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 5.5.36497.1, 10.10.4162596903125.1

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 24 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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
36497Data not computed