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Properties

Label 20.0.64998372267...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{18}\cdot 5^{14}$
Root discriminant $30.95$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100, 0, 560, 0, 2316, 0, 4512, 0, 6420, 0, -558, 0, -696, 0, -132, 0, 60, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 + 60*x^16 - 132*x^14 - 696*x^12 - 558*x^10 + 6420*x^8 + 4512*x^6 + 2316*x^4 + 560*x^2 + 100)
 
gp: K = bnfinit(x^20 + 8*x^18 + 60*x^16 - 132*x^14 - 696*x^12 - 558*x^10 + 6420*x^8 + 4512*x^6 + 2316*x^4 + 560*x^2 + 100, 1)
 

Normalized defining polynomial

\( x^{20} + 8 x^{18} + 60 x^{16} - 132 x^{14} - 696 x^{12} - 558 x^{10} + 6420 x^{8} + 4512 x^{6} + 2316 x^{4} + 560 x^{2} + 100 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(649983722677862400000000000000=2^{38}\cdot 3^{18}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.95$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{7926} a^{16} - \frac{625}{7926} a^{14} + \frac{140}{3963} a^{12} + \frac{371}{7926} a^{10} - \frac{352}{1321} a^{8} - \frac{20}{3963} a^{6} - \frac{636}{1321} a^{4} + \frac{578}{3963} a^{2} + \frac{91}{1321}$, $\frac{1}{7926} a^{17} - \frac{625}{7926} a^{15} + \frac{140}{3963} a^{13} + \frac{371}{7926} a^{11} - \frac{352}{1321} a^{9} - \frac{20}{3963} a^{7} - \frac{636}{1321} a^{5} + \frac{578}{3963} a^{3} + \frac{91}{1321} a$, $\frac{1}{44456283394290} a^{18} - \frac{276786273}{4939587043810} a^{16} + \frac{54185275049}{2963752226286} a^{14} + \frac{636267212681}{14818761131430} a^{12} + \frac{915478703}{14818761131430} a^{10} + \frac{3617280649462}{7409380565715} a^{8} - \frac{50248973621}{1481876113143} a^{6} + \frac{1924042816967}{7409380565715} a^{4} + \frac{831133848192}{2469793521905} a^{2} - \frac{18071890552}{4445628339429}$, $\frac{1}{44456283394290} a^{19} - \frac{276786273}{4939587043810} a^{17} + \frac{54185275049}{2963752226286} a^{15} + \frac{636267212681}{14818761131430} a^{13} + \frac{915478703}{14818761131430} a^{11} + \frac{3617280649462}{7409380565715} a^{9} - \frac{50248973621}{1481876113143} a^{7} + \frac{1924042816967}{7409380565715} a^{5} + \frac{831133848192}{2469793521905} a^{3} - \frac{18071890552}{4445628339429} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{4535638}{1869639305} a^{18} - \frac{35455179}{1869639305} a^{16} - \frac{159657968}{1121783583} a^{14} + \frac{643835466}{1869639305} a^{12} + \frac{6024824001}{3739278610} a^{10} + \frac{6110033012}{5608917915} a^{8} - \frac{5840071692}{373927861} a^{6} - \frac{15073712781}{1869639305} a^{4} - \frac{31366075324}{5608917915} a^{2} - \frac{131163595}{373927861} \) (order $6$)
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:  \( 9660809.80023597 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-3}, \sqrt{10})\), 5.1.162000.1, 10.2.268738560000000.13, 10.0.78732000000.1, 10.0.806215680000000.16

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$