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Properties

Label 20.2.12016571954...7008.3
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{30}\cdot 47^{9}$
Root discriminant $16.00$
Ramified primes $2, 47$
Class number $1$
Class group Trivial
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![1, -8, 44, -138, 396, -642, 788, -786, 728, -668, 540, -414, 269, -168, 124, -60, 26, -18, 6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 - 18*x^17 + 26*x^16 - 60*x^15 + 124*x^14 - 168*x^13 + 269*x^12 - 414*x^11 + 540*x^10 - 668*x^9 + 728*x^8 - 786*x^7 + 788*x^6 - 642*x^5 + 396*x^4 - 138*x^3 + 44*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 6*x^18 - 18*x^17 + 26*x^16 - 60*x^15 + 124*x^14 - 168*x^13 + 269*x^12 - 414*x^11 + 540*x^10 - 668*x^9 + 728*x^8 - 786*x^7 + 788*x^6 - 642*x^5 + 396*x^4 - 138*x^3 + 44*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 6 x^{18} - 18 x^{17} + 26 x^{16} - 60 x^{15} + 124 x^{14} - 168 x^{13} + 269 x^{12} - 414 x^{11} + 540 x^{10} - 668 x^{9} + 728 x^{8} - 786 x^{7} + 788 x^{6} - 642 x^{5} + 396 x^{4} - 138 x^{3} + 44 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1201657195483347978027008=-\,2^{30}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.00$
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]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} - \frac{2}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{1095883132081956545} a^{19} + \frac{13164251166716}{499263385914331} a^{18} - \frac{289274169527181071}{1095883132081956545} a^{17} - \frac{219234898419865439}{1095883132081956545} a^{16} - \frac{19346445386810123}{1095883132081956545} a^{15} + \frac{95590086562040908}{219176626416391309} a^{14} - \frac{492545462261532738}{1095883132081956545} a^{13} + \frac{368040814733037812}{1095883132081956545} a^{12} + \frac{296682971223353271}{1095883132081956545} a^{11} - \frac{54131944239507662}{219176626416391309} a^{10} - \frac{213731549164845873}{1095883132081956545} a^{9} - \frac{90167860008766701}{219176626416391309} a^{8} + \frac{30512891918392392}{1095883132081956545} a^{7} - \frac{304683728465461204}{1095883132081956545} a^{6} + \frac{28844980440055492}{1095883132081956545} a^{5} - \frac{211679414715082881}{1095883132081956545} a^{4} + \frac{225755262537827524}{1095883132081956545} a^{3} + \frac{18574199441555452}{1095883132081956545} a^{2} + \frac{44290283875986229}{1095883132081956545} a - \frac{202812249803795362}{1095883132081956545}$

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:  $10$
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:  \( 7842.51365505 \)
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

5.1.2209.1, 10.2.4996793344.3

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$