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Properties

Label 20.8.76572488767...8161.1
Degree $20$
Signature $[8, 6]$
Discriminant $13^{4}\cdot 401^{9}$
Root discriminant $24.79$
Ramified primes $13, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 14, -47, 49, 124, -550, 1333, -2134, 1936, -620, -842, 1139, -566, -51, 193, -97, 12, 9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 9*x^18 + 12*x^17 - 97*x^16 + 193*x^15 - 51*x^14 - 566*x^13 + 1139*x^12 - 842*x^11 - 620*x^10 + 1936*x^9 - 2134*x^8 + 1333*x^7 - 550*x^6 + 124*x^5 + 49*x^4 - 47*x^3 + 14*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 9*x^18 + 12*x^17 - 97*x^16 + 193*x^15 - 51*x^14 - 566*x^13 + 1139*x^12 - 842*x^11 - 620*x^10 + 1936*x^9 - 2134*x^8 + 1333*x^7 - 550*x^6 + 124*x^5 + 49*x^4 - 47*x^3 + 14*x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 9 x^{18} + 12 x^{17} - 97 x^{16} + 193 x^{15} - 51 x^{14} - 566 x^{13} + 1139 x^{12} - 842 x^{11} - 620 x^{10} + 1936 x^{9} - 2134 x^{8} + 1333 x^{7} - 550 x^{6} + 124 x^{5} + 49 x^{4} - 47 x^{3} + 14 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7657248876721128710150208161=13^{4}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.79$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{2}{9} a^{11} - \frac{1}{9} a^{10} + \frac{2}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{4}{9} a^{10} - \frac{4}{9} a^{9} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{4}{9} a^{11} - \frac{4}{9} a^{10} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{4}{9} a$, $\frac{1}{3199132524053454543} a^{19} + \frac{5471645322311491}{168375396002813397} a^{18} + \frac{7840165898342939}{355459169339272727} a^{17} - \frac{7404536410022048}{168375396002813397} a^{16} - \frac{4133208576798592}{1066377508017818181} a^{15} + \frac{172199771786210083}{1066377508017818181} a^{14} + \frac{32841672452333998}{3199132524053454543} a^{13} - \frac{256526768238784253}{3199132524053454543} a^{12} + \frac{545709793755649625}{3199132524053454543} a^{11} - \frac{1275717203147600318}{3199132524053454543} a^{10} - \frac{1561257699056514170}{3199132524053454543} a^{9} - \frac{1218715692267319399}{3199132524053454543} a^{8} + \frac{1540855950043146659}{3199132524053454543} a^{7} - \frac{156536971040415949}{1066377508017818181} a^{6} - \frac{869548888760851349}{3199132524053454543} a^{5} - \frac{578370861601448963}{3199132524053454543} a^{4} - \frac{1446010553813207576}{3199132524053454543} a^{3} - \frac{1739860996070954}{355459169339272727} a^{2} - \frac{230686409083962131}{3199132524053454543} a + \frac{1087689001441630295}{3199132524053454543}$

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:  $13$
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:  \( 2449529.91298 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

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, 10.6.4369826510569.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$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$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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