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Properties

Label 20.6.37006650106...9375.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,5^{10}\cdot 97^{2}\cdot 3319^{5}$
Root discriminant $26.82$
Ramified primes $5, 97, 3319$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 12, -26, -69, 190, 27, 254, -201, 361, 40, -68, 22, -324, 129, -200, 24, 6, -15, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 12*x^18 - 15*x^17 + 6*x^16 + 24*x^15 - 200*x^14 + 129*x^13 - 324*x^12 + 22*x^11 - 68*x^10 + 40*x^9 + 361*x^8 - 201*x^7 + 254*x^6 + 27*x^5 + 190*x^4 - 69*x^3 - 26*x^2 + 12*x - 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 12*x^18 - 15*x^17 + 6*x^16 + 24*x^15 - 200*x^14 + 129*x^13 - 324*x^12 + 22*x^11 - 68*x^10 + 40*x^9 + 361*x^8 - 201*x^7 + 254*x^6 + 27*x^5 + 190*x^4 - 69*x^3 - 26*x^2 + 12*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 12 x^{18} - 15 x^{17} + 6 x^{16} + 24 x^{15} - 200 x^{14} + 129 x^{13} - 324 x^{12} + 22 x^{11} - 68 x^{10} + 40 x^{9} + 361 x^{8} - 201 x^{7} + 254 x^{6} + 27 x^{5} + 190 x^{4} - 69 x^{3} - 26 x^{2} + 12 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-37006650106723762538974609375=-\,5^{10}\cdot 97^{2}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.82$
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:  $5, 97, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{10055247727335859555540762143} a^{19} - \frac{3229316840761005298514132033}{10055247727335859555540762143} a^{18} - \frac{1250941396875987824094734192}{10055247727335859555540762143} a^{17} + \frac{747926177251728640943502631}{10055247727335859555540762143} a^{16} - \frac{4721122780222003589017038224}{10055247727335859555540762143} a^{15} - \frac{3011950788374449702037370203}{10055247727335859555540762143} a^{14} - \frac{1388394039209100457486413013}{10055247727335859555540762143} a^{13} - \frac{1633625359953618343131670747}{10055247727335859555540762143} a^{12} + \frac{3851016416153398824319300490}{10055247727335859555540762143} a^{11} + \frac{547169377239655930762859752}{3351749242445286518513587381} a^{10} - \frac{4362412470846433598599323581}{10055247727335859555540762143} a^{9} + \frac{2749071883297199464694410076}{10055247727335859555540762143} a^{8} - \frac{1005654042936044627297357714}{3351749242445286518513587381} a^{7} - \frac{567906584663927466750078254}{3351749242445286518513587381} a^{6} + \frac{4378573467686977793810001734}{10055247727335859555540762143} a^{5} - \frac{1586801908635172806144893068}{10055247727335859555540762143} a^{4} - \frac{1365577644016002653638795669}{3351749242445286518513587381} a^{3} - \frac{1132173861571173327557488270}{3351749242445286518513587381} a^{2} + \frac{2052441779846332820989796233}{10055247727335859555540762143} a + \frac{4005611341282514362797743230}{10055247727335859555540762143}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $12$
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:  \( 1890533.3442 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

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 t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.34424253125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
3319Data not computed