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Properties

Label 20.4.96479729228...3024.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{63}\cdot 3^{21}$
Root discriminant $28.13$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -2, 36, -69, 24, 84, -120, -102, 488, -704, 488, -102, -120, 84, 24, -69, 36, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 2*x^18 + 36*x^17 - 69*x^16 + 24*x^15 + 84*x^14 - 120*x^13 - 102*x^12 + 488*x^11 - 704*x^10 + 488*x^9 - 102*x^8 - 120*x^7 + 84*x^6 + 24*x^5 - 69*x^4 + 36*x^3 - 2*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 2*x^18 + 36*x^17 - 69*x^16 + 24*x^15 + 84*x^14 - 120*x^13 - 102*x^12 + 488*x^11 - 704*x^10 + 488*x^9 - 102*x^8 - 120*x^7 + 84*x^6 + 24*x^5 - 69*x^4 + 36*x^3 - 2*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 2 x^{18} + 36 x^{17} - 69 x^{16} + 24 x^{15} + 84 x^{14} - 120 x^{13} - 102 x^{12} + 488 x^{11} - 704 x^{10} + 488 x^{9} - 102 x^{8} - 120 x^{7} + 84 x^{6} + 24 x^{5} - 69 x^{4} + 36 x^{3} - 2 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(96479729228174488169059713024=2^{63}\cdot 3^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.13$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{18} a^{16} - \frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{3} a^{11} + \frac{2}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a + \frac{7}{18}$, $\frac{1}{36} a^{17} - \frac{1}{36} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{2}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{4}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{9} a^{6} + \frac{7}{18} a^{5} - \frac{1}{18} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{7}{36} a + \frac{7}{36}$, $\frac{1}{3452760} a^{18} - \frac{173}{191820} a^{17} + \frac{10393}{383640} a^{16} - \frac{815}{28773} a^{15} + \frac{36919}{575460} a^{14} - \frac{641}{12510} a^{13} + \frac{2569}{38364} a^{12} + \frac{22078}{47955} a^{11} + \frac{439}{47955} a^{10} + \frac{125914}{431595} a^{9} + \frac{439}{47955} a^{8} + \frac{22078}{47955} a^{7} + \frac{2569}{38364} a^{6} - \frac{641}{12510} a^{5} + \frac{36919}{575460} a^{4} - \frac{815}{28773} a^{3} + \frac{10393}{383640} a^{2} - \frac{173}{191820} a + \frac{1}{3452760}$, $\frac{1}{6905520} a^{19} - \frac{1}{6905520} a^{18} - \frac{623}{460368} a^{17} - \frac{33293}{2301840} a^{16} - \frac{128441}{1150920} a^{15} - \frac{64759}{1150920} a^{14} + \frac{130397}{1150920} a^{13} - \frac{174629}{1150920} a^{12} - \frac{50711}{287730} a^{11} - \frac{45439}{431595} a^{10} + \frac{19009}{431595} a^{9} + \frac{1991}{57546} a^{8} + \frac{50041}{383640} a^{7} + \frac{56783}{383640} a^{6} + \frac{64841}{1150920} a^{5} + \frac{12367}{1150920} a^{4} - \frac{43781}{2301840} a^{3} - \frac{257491}{2301840} a^{2} + \frac{1815319}{6905520} a + \frac{770393}{6905520}$

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:  $11$
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:  \( 22770985.3263 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 41 conjugacy class representatives for t20n547
Character table for t20n547 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.13824.1, 10.2.5283615080448.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.15.11$x^{12} - 3 x^{11} + 3 x^{10} + 3 x^{9} + 3 x^{8} - 3 x^{7} + 3 x^{6} - 3 x^{4} - 3$$12$$1$$15$12T42$[3/2]_{4}^{6}$