Properties

Label 20.0.17235683651...5632.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{15}\cdot 47^{10}$
Root discriminant $11.53$
Ramified primes $2, 47$
Class number $1$
Class group Trivial
Galois group $C_5:D_4$ (as 20T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 21, -35, 37, -34, 26, -3, 17, -56, 29, 19, 2, -28, 3, 15, -3, -4, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + x^18 - 4*x^17 - 3*x^16 + 15*x^15 + 3*x^14 - 28*x^13 + 2*x^12 + 19*x^11 + 29*x^10 - 56*x^9 + 17*x^8 - 3*x^7 + 26*x^6 - 34*x^5 + 37*x^4 - 35*x^3 + 21*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + x^{18} - 4 x^{17} - 3 x^{16} + 15 x^{15} + 3 x^{14} - 28 x^{13} + 2 x^{12} + 19 x^{11} + 29 x^{10} - 56 x^{9} + 17 x^{8} - 3 x^{7} + 26 x^{6} - 34 x^{5} + 37 x^{4} - 35 x^{3} + 21 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1723568365103679045632=2^{15}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.53$
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:  $2, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47} a^{18} - \frac{16}{47} a^{17} - \frac{15}{47} a^{16} - \frac{7}{47} a^{15} - \frac{6}{47} a^{14} + \frac{17}{47} a^{13} + \frac{15}{47} a^{12} + \frac{1}{47} a^{11} + \frac{1}{47} a^{10} - \frac{17}{47} a^{9} - \frac{19}{47} a^{8} + \frac{22}{47} a^{7} - \frac{8}{47} a^{6} - \frac{16}{47} a^{5} + \frac{11}{47} a^{4} - \frac{4}{47} a^{3} + \frac{7}{47} a^{2} - \frac{9}{47} a + \frac{9}{47}$, $\frac{1}{24657195037} a^{19} + \frac{14423459}{24657195037} a^{18} + \frac{6052526827}{24657195037} a^{17} + \frac{6076674939}{24657195037} a^{16} + \frac{490195653}{24657195037} a^{15} - \frac{3702211118}{24657195037} a^{14} - \frac{3221954579}{24657195037} a^{13} - \frac{1683844526}{24657195037} a^{12} - \frac{5510042910}{24657195037} a^{11} + \frac{6610333898}{24657195037} a^{10} - \frac{215387826}{524621171} a^{9} + \frac{2552398449}{24657195037} a^{8} - \frac{8641400464}{24657195037} a^{7} + \frac{5276938446}{24657195037} a^{6} + \frac{7916454968}{24657195037} a^{5} - \frac{7803064182}{24657195037} a^{4} - \frac{4935834703}{24657195037} a^{3} + \frac{6673552933}{24657195037} a^{2} - \frac{4891694897}{24657195037} a - \frac{6491582894}{24657195037}$

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:  $9$
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:  \( 143.606119616 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{-47}) \), 4.0.17672.1, 5.1.2209.1 x5, 10.0.229345007.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$