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Properties

Label 20.0.58523882840...8681.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{10}\cdot 41^{12}$
Root discriminant $30.79$
Ramified primes $11, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T145

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1193, -2920, 6082, -12631, 20016, -27669, 32829, -34122, 31800, -25956, 19211, -12391, 7446, -4140, 2276, -1144, 513, -182, 50, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 50*x^18 - 182*x^17 + 513*x^16 - 1144*x^15 + 2276*x^14 - 4140*x^13 + 7446*x^12 - 12391*x^11 + 19211*x^10 - 25956*x^9 + 31800*x^8 - 34122*x^7 + 32829*x^6 - 27669*x^5 + 20016*x^4 - 12631*x^3 + 6082*x^2 - 2920*x + 1193)
 
gp: K = bnfinit(x^20 - 9*x^19 + 50*x^18 - 182*x^17 + 513*x^16 - 1144*x^15 + 2276*x^14 - 4140*x^13 + 7446*x^12 - 12391*x^11 + 19211*x^10 - 25956*x^9 + 31800*x^8 - 34122*x^7 + 32829*x^6 - 27669*x^5 + 20016*x^4 - 12631*x^3 + 6082*x^2 - 2920*x + 1193, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 50 x^{18} - 182 x^{17} + 513 x^{16} - 1144 x^{15} + 2276 x^{14} - 4140 x^{13} + 7446 x^{12} - 12391 x^{11} + 19211 x^{10} - 25956 x^{9} + 31800 x^{8} - 34122 x^{7} + 32829 x^{6} - 27669 x^{5} + 20016 x^{4} - 12631 x^{3} + 6082 x^{2} - 2920 x + 1193 \)

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:  \(585238828401142794165873178681=11^{10}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.79$
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:  $11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4588596039828075109236498781458977} a^{19} + \frac{1101415785249243605312845918646963}{4588596039828075109236498781458977} a^{18} + \frac{2279876883379286256346298924647437}{4588596039828075109236498781458977} a^{17} + \frac{1492119109387689400943862904906026}{4588596039828075109236498781458977} a^{16} + \frac{259202965075044596948562229962592}{4588596039828075109236498781458977} a^{15} + \frac{25708887947566394815739939204}{23775109014653238907961133582689} a^{14} - \frac{2215547780762946644323890296456092}{4588596039828075109236498781458977} a^{13} - \frac{1200749090231278334261275974316854}{4588596039828075109236498781458977} a^{12} + \frac{1787853587120879759984346411532733}{4588596039828075109236498781458977} a^{11} - \frac{2018560835273550590251152663953896}{4588596039828075109236498781458977} a^{10} - \frac{765756583814604887124441163328387}{4588596039828075109236498781458977} a^{9} - \frac{1822631339322532080176986242332016}{4588596039828075109236498781458977} a^{8} - \frac{1531237068149808379578951691718485}{4588596039828075109236498781458977} a^{7} + \frac{2079668109763986076481545642159149}{4588596039828075109236498781458977} a^{6} + \frac{1189924344527828059000471225243336}{4588596039828075109236498781458977} a^{5} - \frac{2034848084510693373172543161066334}{4588596039828075109236498781458977} a^{4} - \frac{2002829321398207722492218261410654}{4588596039828075109236498781458977} a^{3} + \frac{1413112472375230839974317946136696}{4588596039828075109236498781458977} a^{2} - \frac{22066907024296382654161979420082}{655513719975439301319499825922711} a - \frac{343804906943986004115281764914939}{4588596039828075109236498781458977}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 1323941.60611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T145:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for t20n145
Character table for t20n145

Intermediate fields

\(\Q(\sqrt{-11}) \), 10.4.6322388744771.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 siblings: 6.0.31083371.1, 6.4.2237411.1
Degree 10 sibling: 10.4.6322388744771.1
Degree 12 siblings: 12.0.5006007982921.1, Deg 12
Degree 15 siblings: Deg 15, 15.7.14145782123826827881.1
Degree 20 siblings: Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41Data not computed