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Properties

Label 20.0.17590143860...6161.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{8}\cdot 401^{9}$
Root discriminant $23.03$
Ramified primes $3, 401$
Class number $2$
Class group $[2]$
Galois group $C_2^4:D_5$ (as 20T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, -80, 32, 36, 166, -747, 1702, -3013, 4157, -4338, 3616, -2770, 2102, -1457, 893, -525, 293, -133, 43, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32)
 
gp: K = bnfinit(x^20 - 9*x^19 + 43*x^18 - 133*x^17 + 293*x^16 - 525*x^15 + 893*x^14 - 1457*x^13 + 2102*x^12 - 2770*x^11 + 3616*x^10 - 4338*x^9 + 4157*x^8 - 3013*x^7 + 1702*x^6 - 747*x^5 + 166*x^4 + 36*x^3 + 32*x^2 - 80*x + 32, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 43 x^{18} - 133 x^{17} + 293 x^{16} - 525 x^{15} + 893 x^{14} - 1457 x^{13} + 2102 x^{12} - 2770 x^{11} + 3616 x^{10} - 4338 x^{9} + 4157 x^{8} - 3013 x^{7} + 1702 x^{6} - 747 x^{5} + 166 x^{4} + 36 x^{3} + 32 x^{2} - 80 x + 32 \)

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:  \(1759014386056767111350986161=3^{8}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.03$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} - \frac{1}{12} a^{15} - \frac{1}{12} a^{14} + \frac{5}{12} a^{13} - \frac{5}{12} a^{12} - \frac{1}{4} a^{11} - \frac{5}{12} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{4} a^{5} - \frac{5}{12} a^{4} - \frac{1}{2} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{1992} a^{18} + \frac{25}{664} a^{17} - \frac{23}{664} a^{16} - \frac{39}{664} a^{15} - \frac{35}{1992} a^{14} - \frac{87}{664} a^{13} + \frac{205}{1992} a^{12} + \frac{743}{1992} a^{11} - \frac{149}{996} a^{10} - \frac{247}{996} a^{9} - \frac{31}{249} a^{8} + \frac{155}{996} a^{7} + \frac{71}{664} a^{6} + \frac{189}{664} a^{5} - \frac{63}{332} a^{4} - \frac{277}{664} a^{3} + \frac{109}{996} a^{2} - \frac{59}{249} a - \frac{67}{249}$, $\frac{1}{56725279539304272} a^{19} - \frac{2593601285403}{18908426513101424} a^{18} - \frac{131517172248745}{56725279539304272} a^{17} - \frac{1876829516645041}{56725279539304272} a^{16} - \frac{209120805001409}{1454494347161648} a^{15} + \frac{2327284266290445}{18908426513101424} a^{14} + \frac{10644646760951033}{56725279539304272} a^{13} + \frac{19491208500406835}{56725279539304272} a^{12} + \frac{1817213763976371}{9454213256550712} a^{11} + \frac{8748872578535917}{28362639769652136} a^{10} - \frac{1770610973298835}{7090659942413034} a^{9} + \frac{317566849292689}{9454213256550712} a^{8} - \frac{14080178203132339}{56725279539304272} a^{7} - \frac{3229224656053157}{56725279539304272} a^{6} + \frac{3381794491492139}{9454213256550712} a^{5} - \frac{26618239103068367}{56725279539304272} a^{4} + \frac{3369051399498763}{28362639769652136} a^{3} - \frac{843954194227}{6571510604646} a^{2} + \frac{1008226401858699}{2363553314137678} a - \frac{262575187935147}{1181776657068839}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 110300.954595 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.2094413889681.1

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 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed