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Properties

Label 20.4.87960930222...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{55}\cdot 5^{12}$
Root discriminant $17.67$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, -10, 4, 0, 40, -40, 40, -90, -40, -40, -40, 0, -4, -10, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^16 - 4*x^15 - 40*x^13 - 40*x^12 - 40*x^11 - 90*x^10 + 40*x^9 - 40*x^8 + 40*x^7 + 4*x^5 - 10*x^4 + 1)
 
gp: K = bnfinit(x^20 - 10*x^16 - 4*x^15 - 40*x^13 - 40*x^12 - 40*x^11 - 90*x^10 + 40*x^9 - 40*x^8 + 40*x^7 + 4*x^5 - 10*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{16} - 4 x^{15} - 40 x^{13} - 40 x^{12} - 40 x^{11} - 90 x^{10} + 40 x^{9} - 40 x^{8} + 40 x^{7} + 4 x^{5} - 10 x^{4} + 1 \)

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:  \(8796093022208000000000000=2^{55}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.67$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{2}{5} a^{12} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{123717670} a^{18} - \frac{906592}{12371767} a^{17} + \frac{11636193}{123717670} a^{16} - \frac{1709023}{3638755} a^{15} + \frac{51002981}{123717670} a^{14} - \frac{988131}{61858835} a^{13} - \frac{34544777}{123717670} a^{12} + \frac{17854419}{61858835} a^{11} - \frac{16188271}{123717670} a^{10} - \frac{19673512}{61858835} a^{9} - \frac{58042331}{123717670} a^{8} + \frac{30226186}{61858835} a^{7} - \frac{7937165}{24743534} a^{6} - \frac{988131}{61858835} a^{5} + \frac{9594231}{24743534} a^{4} - \frac{981272}{3638755} a^{3} + \frac{7570175}{24743534} a^{2} + \frac{7838807}{61858835} a + \frac{24743533}{123717670}$, $\frac{1}{247435340} a^{19} - \frac{1}{247435340} a^{18} - \frac{15677613}{247435340} a^{17} + \frac{4041421}{247435340} a^{16} - \frac{103461773}{247435340} a^{15} + \frac{39351437}{247435340} a^{14} + \frac{77722777}{247435340} a^{13} + \frac{6513695}{49487068} a^{12} + \frac{3976947}{247435340} a^{11} - \frac{47077067}{247435340} a^{10} - \frac{5265681}{49487068} a^{9} - \frac{105022323}{247435340} a^{8} + \frac{4673737}{14555020} a^{7} + \frac{75394703}{247435340} a^{6} + \frac{36521039}{247435340} a^{5} + \frac{73770257}{247435340} a^{4} + \frac{106077927}{247435340} a^{3} + \frac{27760013}{247435340} a^{2} - \frac{27313807}{247435340} a - \frac{33809453}{247435340}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 58377.9345442 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

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 $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 5.1.256000.1, 10.2.524288000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
5Data not computed