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Properties

Label 20.12.3601957905...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 19^{6}\cdot 97^{2}\cdot 1699^{4}$
Root discriminant $37.83$
Ramified primes $5, 19, 97, 1699$
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![-89, -458, 685, 6048, 4488, -11937, -22667, -28237, -9493, 13582, 9296, 1284, -965, -33, 299, -263, -49, 59, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 6*x^18 + 59*x^17 - 49*x^16 - 263*x^15 + 299*x^14 - 33*x^13 - 965*x^12 + 1284*x^11 + 9296*x^10 + 13582*x^9 - 9493*x^8 - 28237*x^7 - 22667*x^6 - 11937*x^5 + 4488*x^4 + 6048*x^3 + 685*x^2 - 458*x - 89)
 
gp: K = bnfinit(x^20 - 4*x^19 - 6*x^18 + 59*x^17 - 49*x^16 - 263*x^15 + 299*x^14 - 33*x^13 - 965*x^12 + 1284*x^11 + 9296*x^10 + 13582*x^9 - 9493*x^8 - 28237*x^7 - 22667*x^6 - 11937*x^5 + 4488*x^4 + 6048*x^3 + 685*x^2 - 458*x - 89, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 6 x^{18} + 59 x^{17} - 49 x^{16} - 263 x^{15} + 299 x^{14} - 33 x^{13} - 965 x^{12} + 1284 x^{11} + 9296 x^{10} + 13582 x^{9} - 9493 x^{8} - 28237 x^{7} - 22667 x^{6} - 11937 x^{5} + 4488 x^{4} + 6048 x^{3} + 685 x^{2} - 458 x - 89 \)

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:  \(36019579053467556544112587890625=5^{10}\cdot 19^{6}\cdot 97^{2}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.83$
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:  $5, 19, 97, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{833420603897550230961861252529993875147323} a^{19} - \frac{276814659023974550007813314964620472152205}{833420603897550230961861252529993875147323} a^{18} - \frac{55963481861441707642363772638197733485964}{833420603897550230961861252529993875147323} a^{17} - \frac{106311297366013446547317927973205954633234}{833420603897550230961861252529993875147323} a^{16} + \frac{7667536510776370755660623983683977781457}{833420603897550230961861252529993875147323} a^{15} - \frac{288880694434716357441193203036981355705602}{833420603897550230961861252529993875147323} a^{14} - \frac{48357839367249094942344326067458314075133}{833420603897550230961861252529993875147323} a^{13} - \frac{43735426160167893114476791783216945152388}{833420603897550230961861252529993875147323} a^{12} + \frac{46304101072671857300644790308125370082915}{833420603897550230961861252529993875147323} a^{11} - \frac{356162042788262058884069955239170855897483}{833420603897550230961861252529993875147323} a^{10} - \frac{82291962462318675712216600983642407440991}{833420603897550230961861252529993875147323} a^{9} + \frac{89017792978649184717005337203667621978442}{833420603897550230961861252529993875147323} a^{8} + \frac{42312273665772021170024594018760205948058}{833420603897550230961861252529993875147323} a^{7} - \frac{204096565458873994748226596895910546687879}{833420603897550230961861252529993875147323} a^{6} - \frac{190693858891839757661113608783517166060340}{833420603897550230961861252529993875147323} a^{5} - \frac{400708040864410702777834484237136669971061}{833420603897550230961861252529993875147323} a^{4} + \frac{249973393909914849434871735154054178961838}{833420603897550230961861252529993875147323} a^{3} + \frac{116137707171787957021812990224073363125688}{833420603897550230961861252529993875147323} a^{2} - \frac{49046718085479311210624412983233262830638}{833420603897550230961861252529993875147323} a - \frac{399848759356067696286209377238572286088941}{833420603897550230961861252529993875147323}$

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:  \( 323360926.193 \) (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

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$97$97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
1699Data not computed