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Properties

Label 20.8.18981351240...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{28}\cdot 5^{11}\cdot 3469^{4}$
Root discriminant $32.65$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T755

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 88, 184, -80, 536, -3356, 3980, -4556, 7604, -7044, 5086, -4900, 3568, -1728, 990, -586, 208, -56, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 24*x^18 - 56*x^17 + 208*x^16 - 586*x^15 + 990*x^14 - 1728*x^13 + 3568*x^12 - 4900*x^11 + 5086*x^10 - 7044*x^9 + 7604*x^8 - 4556*x^7 + 3980*x^6 - 3356*x^5 + 536*x^4 - 80*x^3 + 184*x^2 + 88*x + 4)
 
gp: K = bnfinit(x^20 - 8*x^19 + 24*x^18 - 56*x^17 + 208*x^16 - 586*x^15 + 990*x^14 - 1728*x^13 + 3568*x^12 - 4900*x^11 + 5086*x^10 - 7044*x^9 + 7604*x^8 - 4556*x^7 + 3980*x^6 - 3356*x^5 + 536*x^4 - 80*x^3 + 184*x^2 + 88*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 24 x^{18} - 56 x^{17} + 208 x^{16} - 586 x^{15} + 990 x^{14} - 1728 x^{13} + 3568 x^{12} - 4900 x^{11} + 5086 x^{10} - 7044 x^{9} + 7604 x^{8} - 4556 x^{7} + 3980 x^{6} - 3356 x^{5} + 536 x^{4} - 80 x^{3} + 184 x^{2} + 88 x + 4 \)

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:  \(1898135124043641651200000000000=2^{28}\cdot 5^{11}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.65$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{62} a^{17} - \frac{3}{62} a^{16} - \frac{11}{62} a^{15} - \frac{1}{62} a^{14} - \frac{3}{31} a^{13} + \frac{1}{62} a^{12} + \frac{11}{62} a^{11} - \frac{9}{62} a^{10} - \frac{13}{31} a^{9} - \frac{11}{31} a^{8} + \frac{12}{31} a^{7} + \frac{14}{31} a^{6} + \frac{13}{31} a^{5} - \frac{2}{31} a^{4} + \frac{2}{31} a^{3} + \frac{14}{31} a^{2} + \frac{12}{31} a - \frac{5}{31}$, $\frac{1}{62} a^{18} + \frac{11}{62} a^{16} - \frac{3}{62} a^{15} - \frac{9}{62} a^{14} + \frac{7}{31} a^{13} + \frac{7}{31} a^{12} - \frac{7}{62} a^{11} + \frac{9}{62} a^{10} + \frac{12}{31} a^{9} + \frac{10}{31} a^{8} - \frac{12}{31} a^{7} - \frac{7}{31} a^{6} + \frac{6}{31} a^{5} - \frac{4}{31} a^{4} - \frac{11}{31} a^{3} - \frac{8}{31} a^{2} - \frac{15}{31}$, $\frac{1}{129323849904108334} a^{19} + \frac{602058972510199}{129323849904108334} a^{18} - \frac{413338187509715}{129323849904108334} a^{17} + \frac{24803475266232867}{129323849904108334} a^{16} - \frac{25406730363989445}{129323849904108334} a^{15} - \frac{7840568632139284}{64661924952054167} a^{14} + \frac{30141523274906537}{129323849904108334} a^{13} + \frac{19910448422012111}{129323849904108334} a^{12} - \frac{13296998712947411}{64661924952054167} a^{11} + \frac{8783273459470124}{64661924952054167} a^{10} + \frac{28209967954749227}{64661924952054167} a^{9} + \frac{25460264494254983}{64661924952054167} a^{8} + \frac{3471165095659444}{64661924952054167} a^{7} + \frac{6724896767642879}{64661924952054167} a^{6} - \frac{9336668004670557}{64661924952054167} a^{5} - \frac{29733940364639079}{64661924952054167} a^{4} + \frac{30534186106509949}{64661924952054167} a^{3} - \frac{4238846153725419}{64661924952054167} a^{2} + \frac{16897378813868178}{64661924952054167} a - \frac{17183042796305233}{64661924952054167}$

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:  \( 40977743.8647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T755:

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 t20n755 are not computed
Character table for t20n755 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed