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Properties

Label 20.8.21240350936...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{32}\cdot 5^{12}\cdot 1193^{4}$
Root discriminant $32.84$
Ramified primes $2, 5, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, -192, 384, -160, -648, 1440, -1160, -488, 2186, -1988, 414, 428, -315, 52, 18, 28, -33, 8, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 4*x^18 + 8*x^17 - 33*x^16 + 28*x^15 + 18*x^14 + 52*x^13 - 315*x^12 + 428*x^11 + 414*x^10 - 1988*x^9 + 2186*x^8 - 488*x^7 - 1160*x^6 + 1440*x^5 - 648*x^4 - 160*x^3 + 384*x^2 - 192*x + 32)
 
gp: K = bnfinit(x^20 - 4*x^19 + 4*x^18 + 8*x^17 - 33*x^16 + 28*x^15 + 18*x^14 + 52*x^13 - 315*x^12 + 428*x^11 + 414*x^10 - 1988*x^9 + 2186*x^8 - 488*x^7 - 1160*x^6 + 1440*x^5 - 648*x^4 - 160*x^3 + 384*x^2 - 192*x + 32, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 4 x^{18} + 8 x^{17} - 33 x^{16} + 28 x^{15} + 18 x^{14} + 52 x^{13} - 315 x^{12} + 428 x^{11} + 414 x^{10} - 1988 x^{9} + 2186 x^{8} - 488 x^{7} - 1160 x^{6} + 1440 x^{5} - 648 x^{4} - 160 x^{3} + 384 x^{2} - 192 x + 32 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2124035093693464576000000000000=2^{32}\cdot 5^{12}\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.84$
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:  $2, 5, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{14} - \frac{1}{16} a^{12} - \frac{1}{4} a^{9} + \frac{1}{16} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{15} - \frac{1}{16} a^{13} - \frac{1}{4} a^{10} + \frac{1}{16} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{18} + \frac{3}{32} a^{14} - \frac{1}{4} a^{13} - \frac{1}{16} a^{12} + \frac{1}{8} a^{11} - \frac{7}{32} a^{10} - \frac{3}{16} a^{8} + \frac{3}{8} a^{7} - \frac{5}{16} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{1595407977585040480} a^{19} - \frac{655828445044371}{49856499299532515} a^{18} + \frac{3085020047833993}{159540797758504048} a^{17} - \frac{6727512297073463}{398851994396260120} a^{16} - \frac{40897518798256937}{1595407977585040480} a^{15} - \frac{9323017674251527}{199425997198130060} a^{14} - \frac{73084697376424551}{398851994396260120} a^{13} + \frac{8837623839512527}{49856499299532515} a^{12} - \frac{233498542961034047}{1595407977585040480} a^{11} + \frac{3182020808083362}{49856499299532515} a^{10} - \frac{6006745904460427}{398851994396260120} a^{9} - \frac{2650345367942407}{49856499299532515} a^{8} + \frac{201904361328127559}{797703988792520240} a^{7} - \frac{1516389718125571}{49856499299532515} a^{6} + \frac{63809904686217359}{398851994396260120} a^{5} + \frac{38211523558730439}{199425997198130060} a^{4} - \frac{86237671889205123}{199425997198130060} a^{3} - \frac{15223300580833764}{49856499299532515} a^{2} - \frac{12164533456178101}{49856499299532515} a + \frac{17132624347450602}{49856499299532515}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 97261846.6711 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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

Sibling fields

Degree 20 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
1193Data not computed