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Properties

Label 20.8.32224748018...5824.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{36}\cdot 97^{2}\cdot 2657^{4}$
Root discriminant $26.63$
Ramified primes $2, 97, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -9, -16, 53, 60, -313, -192, 1302, -232, -2753, 2840, 326, -1980, 919, 220, -309, 74, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 12*x^18 + 74*x^17 - 309*x^16 + 220*x^15 + 919*x^14 - 1980*x^13 + 326*x^12 + 2840*x^11 - 2753*x^10 - 232*x^9 + 1302*x^8 - 192*x^7 - 313*x^6 + 60*x^5 + 53*x^4 - 16*x^3 - 9*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 12*x^18 + 74*x^17 - 309*x^16 + 220*x^15 + 919*x^14 - 1980*x^13 + 326*x^12 + 2840*x^11 - 2753*x^10 - 232*x^9 + 1302*x^8 - 192*x^7 - 313*x^6 + 60*x^5 + 53*x^4 - 16*x^3 - 9*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 12 x^{18} + 74 x^{17} - 309 x^{16} + 220 x^{15} + 919 x^{14} - 1980 x^{13} + 326 x^{12} + 2840 x^{11} - 2753 x^{10} - 232 x^{9} + 1302 x^{8} - 192 x^{7} - 313 x^{6} + 60 x^{5} + 53 x^{4} - 16 x^{3} - 9 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(32224748018872701886161485824=2^{36}\cdot 97^{2}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.63$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{16} - \frac{2}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{27168653721395} a^{19} - \frac{1605640766843}{27168653721395} a^{18} - \frac{7583054932852}{27168653721395} a^{17} - \frac{12347481022371}{27168653721395} a^{16} + \frac{1085186272789}{27168653721395} a^{15} + \frac{11163938938918}{27168653721395} a^{14} + \frac{7735433588772}{27168653721395} a^{13} + \frac{11401138245337}{27168653721395} a^{12} + \frac{2090378795414}{27168653721395} a^{11} - \frac{11791961830644}{27168653721395} a^{10} - \frac{13562230509751}{27168653721395} a^{9} + \frac{2230217020424}{5433730744279} a^{8} - \frac{10097570812583}{27168653721395} a^{7} + \frac{3150954016063}{27168653721395} a^{6} + \frac{6299424901879}{27168653721395} a^{5} - \frac{8686341189906}{27168653721395} a^{4} - \frac{3116471893221}{27168653721395} a^{3} - \frac{4580219182864}{27168653721395} a^{2} - \frac{3550948498799}{27168653721395} a - \frac{3623256573547}{27168653721395}$

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:  \( 9703010.17278 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

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 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.12.24.336$x^{12} + 4 x^{11} - 2 x^{10} + 4 x^{6} + 4 x^{5} + 4 x^{4} - 2 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[4/3, 4/3, 3]_{3}^{2}$
97Data not computed
2657Data not computed