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Properties

Label 20.4.32271198507...5609.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{6}\cdot 401^{8}$
Root discriminant $23.74$
Ramified primes $13, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 52, -779, 2797, -4787, 2834, 6032, -19118, 29472, -31545, 25692, -16718, 9061, -4404, 2090, -1004, 457, -174, 51, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 51*x^18 - 174*x^17 + 457*x^16 - 1004*x^15 + 2090*x^14 - 4404*x^13 + 9061*x^12 - 16718*x^11 + 25692*x^10 - 31545*x^9 + 29472*x^8 - 19118*x^7 + 6032*x^6 + 2834*x^5 - 4787*x^4 + 2797*x^3 - 779*x^2 + 52*x + 13)
 
gp: K = bnfinit(x^20 - 10*x^19 + 51*x^18 - 174*x^17 + 457*x^16 - 1004*x^15 + 2090*x^14 - 4404*x^13 + 9061*x^12 - 16718*x^11 + 25692*x^10 - 31545*x^9 + 29472*x^8 - 19118*x^7 + 6032*x^6 + 2834*x^5 - 4787*x^4 + 2797*x^3 - 779*x^2 + 52*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 51 x^{18} - 174 x^{17} + 457 x^{16} - 1004 x^{15} + 2090 x^{14} - 4404 x^{13} + 9061 x^{12} - 16718 x^{11} + 25692 x^{10} - 31545 x^{9} + 29472 x^{8} - 19118 x^{7} + 6032 x^{6} + 2834 x^{5} - 4787 x^{4} + 2797 x^{3} - 779 x^{2} + 52 x + 13 \)

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:  \(3227119850787707611010935609=13^{6}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.74$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1795610031261} a^{18} - \frac{3}{598536677087} a^{17} - \frac{27652140571}{598536677087} a^{16} + \frac{221217124636}{598536677087} a^{15} + \frac{819260040920}{1795610031261} a^{14} + \frac{607380982066}{1795610031261} a^{13} + \frac{34503591520}{598536677087} a^{12} + \frac{131799726116}{1795610031261} a^{11} - \frac{273549177074}{1795610031261} a^{10} + \frac{192334195946}{1795610031261} a^{9} - \frac{7452890528}{1795610031261} a^{8} + \frac{378772349036}{1795610031261} a^{7} + \frac{222175584707}{1795610031261} a^{6} - \frac{429076574306}{1795610031261} a^{5} + \frac{826283853476}{1795610031261} a^{4} + \frac{268935289159}{1795610031261} a^{3} + \frac{448780269017}{1795610031261} a^{2} - \frac{278629312760}{1795610031261} a - \frac{501228227236}{1795610031261}$, $\frac{1}{1100708949162993} a^{19} + \frac{99}{366902983054331} a^{18} - \frac{100038581497996}{1100708949162993} a^{17} - \frac{92355658181101}{1100708949162993} a^{16} + \frac{519923945958704}{1100708949162993} a^{15} - \frac{417264514802593}{1100708949162993} a^{14} - \frac{393421412133460}{1100708949162993} a^{13} - \frac{153141736478407}{1100708949162993} a^{12} - \frac{290933615414689}{1100708949162993} a^{11} - \frac{398942542813547}{1100708949162993} a^{10} + \frac{380261006664667}{1100708949162993} a^{9} + \frac{429643132027195}{1100708949162993} a^{8} + \frac{41502009289647}{366902983054331} a^{7} + \frac{87308362689907}{1100708949162993} a^{6} + \frac{350752340493788}{1100708949162993} a^{5} + \frac{64219863355676}{366902983054331} a^{4} - \frac{8403388182556}{57932049955947} a^{3} + \frac{21861457255363}{57932049955947} a^{2} - \frac{155676553745405}{366902983054331} a - \frac{116964313657642}{366902983054331}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 729619.578048 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

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

5.5.160801.1, 10.6.56807744637397.1, 10.6.336140500813.1, 10.2.4369826510569.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed