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Properties

Label 20.16.1865598674...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{18}$
Root discriminant $36.61$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T130

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, 844, -2176, -4286, 17475, -5428, -31670, 36118, 2236, -28072, 17040, 3102, -8147, 2760, 976, -908, 105, 90, -26, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 26*x^18 + 90*x^17 + 105*x^16 - 908*x^15 + 976*x^14 + 2760*x^13 - 8147*x^12 + 3102*x^11 + 17040*x^10 - 28072*x^9 + 2236*x^8 + 36118*x^7 - 31670*x^6 - 5428*x^5 + 17475*x^4 - 4286*x^3 - 2176*x^2 + 844*x - 43)
 
gp: K = bnfinit(x^20 - 2*x^19 - 26*x^18 + 90*x^17 + 105*x^16 - 908*x^15 + 976*x^14 + 2760*x^13 - 8147*x^12 + 3102*x^11 + 17040*x^10 - 28072*x^9 + 2236*x^8 + 36118*x^7 - 31670*x^6 - 5428*x^5 + 17475*x^4 - 4286*x^3 - 2176*x^2 + 844*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 26 x^{18} + 90 x^{17} + 105 x^{16} - 908 x^{15} + 976 x^{14} + 2760 x^{13} - 8147 x^{12} + 3102 x^{11} + 17040 x^{10} - 28072 x^{9} + 2236 x^{8} + 36118 x^{7} - 31670 x^{6} - 5428 x^{5} + 17475 x^{4} - 4286 x^{3} - 2176 x^{2} + 844 x - 43 \)

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:  \(18655986742119776375747379200000=2^{30}\cdot 5^{5}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.61$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1732733963003294340253683913295441} a^{19} + \frac{437943763860590993315011150023215}{1732733963003294340253683913295441} a^{18} - \frac{299383484717869661120781998821873}{1732733963003294340253683913295441} a^{17} + \frac{203643596772865963241436828658386}{1732733963003294340253683913295441} a^{16} + \frac{683520002149020648917768910350109}{1732733963003294340253683913295441} a^{15} - \frac{635091487617286111130917171058341}{1732733963003294340253683913295441} a^{14} - \frac{280251780822054295993672097231121}{1732733963003294340253683913295441} a^{13} + \frac{215498654294844166580841459451586}{1732733963003294340253683913295441} a^{12} + \frac{354607369741327114597497132420826}{1732733963003294340253683913295441} a^{11} + \frac{253855902907396749385428446919903}{1732733963003294340253683913295441} a^{10} + \frac{55900353130075150003723748934566}{1732733963003294340253683913295441} a^{9} - \frac{439203634480749936401090392467612}{1732733963003294340253683913295441} a^{8} + \frac{680496814641769961016631232305564}{1732733963003294340253683913295441} a^{7} - \frac{200648984448604546103532643549400}{1732733963003294340253683913295441} a^{6} - \frac{368358348308730288521071982616592}{1732733963003294340253683913295441} a^{5} + \frac{526748316281686524597130447584625}{1732733963003294340253683913295441} a^{4} - \frac{347280599828539178337823589547019}{1732733963003294340253683913295441} a^{3} + \frac{8130113611645737754897926897169}{40296138674495217215201951471987} a^{2} + \frac{810981504978399890133234900514753}{1732733963003294340253683913295441} a + \frac{723122142294890672375396443255}{40296138674495217215201951471987}$

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:  \( 679541499.106 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$