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Properties

Label 20.12.7009755589...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{4}\cdot 3^{10}\cdot 5^{10}\cdot 52501^{4}$
Root discriminant $39.11$
Ramified primes $2, 3, 5, 52501$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T760

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1991, 1424, 13230, -4598, -27964, 10399, 28792, -14415, -15941, 11066, 4636, -4737, -659, 1225, 60, -248, 2, 44, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 6*x^18 + 44*x^17 + 2*x^16 - 248*x^15 + 60*x^14 + 1225*x^13 - 659*x^12 - 4737*x^11 + 4636*x^10 + 11066*x^9 - 15941*x^8 - 14415*x^7 + 28792*x^6 + 10399*x^5 - 27964*x^4 - 4598*x^3 + 13230*x^2 + 1424*x - 1991)
 
gp: K = bnfinit(x^20 - 4*x^19 - 6*x^18 + 44*x^17 + 2*x^16 - 248*x^15 + 60*x^14 + 1225*x^13 - 659*x^12 - 4737*x^11 + 4636*x^10 + 11066*x^9 - 15941*x^8 - 14415*x^7 + 28792*x^6 + 10399*x^5 - 27964*x^4 - 4598*x^3 + 13230*x^2 + 1424*x - 1991, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 6 x^{18} + 44 x^{17} + 2 x^{16} - 248 x^{15} + 60 x^{14} + 1225 x^{13} - 659 x^{12} - 4737 x^{11} + 4636 x^{10} + 11066 x^{9} - 15941 x^{8} - 14415 x^{7} + 28792 x^{6} + 10399 x^{5} - 27964 x^{4} - 4598 x^{3} + 13230 x^{2} + 1424 x - 1991 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(70097555898814099663913906250000=2^{4}\cdot 3^{10}\cdot 5^{10}\cdot 52501^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.11$
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:  $2, 3, 5, 52501$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{3356} a^{18} - \frac{325}{3356} a^{17} - \frac{397}{3356} a^{16} - \frac{691}{1678} a^{15} - \frac{311}{839} a^{14} - \frac{52}{839} a^{13} + \frac{755}{3356} a^{12} + \frac{495}{1678} a^{11} + \frac{439}{3356} a^{10} + \frac{847}{3356} a^{9} + \frac{277}{1678} a^{8} - \frac{213}{3356} a^{7} - \frac{217}{1678} a^{6} - \frac{629}{1678} a^{5} + \frac{317}{3356} a^{4} + \frac{1431}{3356} a^{3} + \frac{207}{3356} a^{2} + \frac{215}{1678} a - \frac{109}{839}$, $\frac{1}{20559337692019396} a^{19} + \frac{2093461233071}{20559337692019396} a^{18} + \frac{32072711356317}{20559337692019396} a^{17} - \frac{515420186389983}{10279668846009698} a^{16} + \frac{2363979778279030}{5139834423004849} a^{15} - \frac{1921825054174739}{10279668846009698} a^{14} - \frac{4684716083976645}{20559337692019396} a^{13} + \frac{3773187238236567}{10279668846009698} a^{12} - \frac{3542926580318543}{20559337692019396} a^{11} - \frac{1679276922833715}{20559337692019396} a^{10} - \frac{1314538661365668}{5139834423004849} a^{9} - \frac{7155810052991809}{20559337692019396} a^{8} + \frac{679197856334487}{5139834423004849} a^{7} + \frac{287556345995669}{5139834423004849} a^{6} - \frac{4903853367467117}{20559337692019396} a^{5} - \frac{2428914085226417}{20559337692019396} a^{4} + \frac{10082566231974353}{20559337692019396} a^{3} - \frac{243048659961523}{10279668846009698} a^{2} - \frac{241670253088584}{5139834423004849} a + \frac{3198382222043923}{10279668846009698}$

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:  $15$
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:  \( 673170267.007 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T760:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n760 are not computed
Character table for t20n760 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8613609378125.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
3Data not computed
5Data not computed
52501Data not computed