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Properties

Label 20.4.30199201883...8809.1
Degree $20$
Signature $[4, 8]$
Discriminant $53^{2}\cdot 401^{10}$
Root discriminant $29.79$
Ramified primes $53, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T81)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5041, -27619, 63448, -72425, 21212, 65930, -130431, 138976, -103758, 57596, -23714, 6861, -1045, -264, 378, -302, 193, -95, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 34*x^18 - 95*x^17 + 193*x^16 - 302*x^15 + 378*x^14 - 264*x^13 - 1045*x^12 + 6861*x^11 - 23714*x^10 + 57596*x^9 - 103758*x^8 + 138976*x^7 - 130431*x^6 + 65930*x^5 + 21212*x^4 - 72425*x^3 + 63448*x^2 - 27619*x + 5041)
 
gp: K = bnfinit(x^20 - 8*x^19 + 34*x^18 - 95*x^17 + 193*x^16 - 302*x^15 + 378*x^14 - 264*x^13 - 1045*x^12 + 6861*x^11 - 23714*x^10 + 57596*x^9 - 103758*x^8 + 138976*x^7 - 130431*x^6 + 65930*x^5 + 21212*x^4 - 72425*x^3 + 63448*x^2 - 27619*x + 5041, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 34 x^{18} - 95 x^{17} + 193 x^{16} - 302 x^{15} + 378 x^{14} - 264 x^{13} - 1045 x^{12} + 6861 x^{11} - 23714 x^{10} + 57596 x^{9} - 103758 x^{8} + 138976 x^{7} - 130431 x^{6} + 65930 x^{5} + 21212 x^{4} - 72425 x^{3} + 63448 x^{2} - 27619 x + 5041 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(301992018836125131097356038809=53^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.79$
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:  $53, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{17} + \frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{2}{9} a^{12} - \frac{2}{9} a^{11} + \frac{2}{9} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{1359614575012748646882631278034359} a^{19} + \frac{17770694606137413684587788549756}{1359614575012748646882631278034359} a^{18} - \frac{170682184434808550651858253587590}{1359614575012748646882631278034359} a^{17} + \frac{1517922651897672963825991685651}{453204858337582882294210426011453} a^{16} - \frac{41373305521802435484335719832761}{1359614575012748646882631278034359} a^{15} + \frac{69618233412456801676503413969747}{1359614575012748646882631278034359} a^{14} - \frac{3570359682301327238926754355023}{453204858337582882294210426011453} a^{13} - \frac{114840036240174642894268953783808}{1359614575012748646882631278034359} a^{12} - \frac{504624777039520558318578849715241}{1359614575012748646882631278034359} a^{11} - \frac{508053351158947240752924893294267}{1359614575012748646882631278034359} a^{10} + \frac{854989706607352131701517359708}{2127722339613065175090189793481} a^{9} + \frac{89710063635683822823683883629168}{1359614575012748646882631278034359} a^{8} + \frac{43628987444415893791053124142189}{151068286112527627431403475337151} a^{7} + \frac{41920507174959442458595481343345}{151068286112527627431403475337151} a^{6} - \frac{311808987376798396980176963821858}{1359614575012748646882631278034359} a^{5} - \frac{75232318962793751232064240720589}{1359614575012748646882631278034359} a^{4} - \frac{129834830913025483253612785356445}{453204858337582882294210426011453} a^{3} - \frac{22968359267587071111086420711121}{151068286112527627431403475337151} a^{2} + \frac{33647782365818664562372058313776}{453204858337582882294210426011453} a - \frac{1633176638126423899718310085840}{19149501056517586575811708141329}$

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:  $11$
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:  \( 7503240.50774 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T81):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.2.1370418964853.1, 10.2.549538004906053.1, 10.10.10368641602001.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed