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Properties

Label 20.0.17374741604...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{5}\cdot 11^{18}$
Root discriminant $12.94$
Ramified primes $5, 11$
Class number $1$
Class group Trivial
Galois group $C_5\times D_4$ (as 20T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 15, -19, 8, 21, -63, 80, -15, -116, 230, -291, 304, -267, 201, -131, 74, -36, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 36*x^17 + 74*x^16 - 131*x^15 + 201*x^14 - 267*x^13 + 304*x^12 - 291*x^11 + 230*x^10 - 116*x^9 - 15*x^8 + 80*x^7 - 63*x^6 + 21*x^5 + 8*x^4 - 19*x^3 + 15*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 36*x^17 + 74*x^16 - 131*x^15 + 201*x^14 - 267*x^13 + 304*x^12 - 291*x^11 + 230*x^10 - 116*x^9 - 15*x^8 + 80*x^7 - 63*x^6 + 21*x^5 + 8*x^4 - 19*x^3 + 15*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 15 x^{18} - 36 x^{17} + 74 x^{16} - 131 x^{15} + 201 x^{14} - 267 x^{13} + 304 x^{12} - 291 x^{11} + 230 x^{10} - 116 x^{9} - 15 x^{8} + 80 x^{7} - 63 x^{6} + 21 x^{5} + 8 x^{4} - 19 x^{3} + 15 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17374741604663223378125=5^{5}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.94$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2069} a^{18} - \frac{914}{2069} a^{17} + \frac{371}{2069} a^{16} - \frac{340}{2069} a^{15} - \frac{451}{2069} a^{14} - \frac{602}{2069} a^{13} + \frac{379}{2069} a^{12} + \frac{870}{2069} a^{11} + \frac{396}{2069} a^{10} + \frac{134}{2069} a^{9} - \frac{144}{2069} a^{8} + \frac{687}{2069} a^{7} - \frac{180}{2069} a^{6} + \frac{316}{2069} a^{5} - \frac{366}{2069} a^{4} + \frac{977}{2069} a^{3} + \frac{953}{2069} a^{2} + \frac{120}{2069} a + \frac{700}{2069}$, $\frac{1}{2069} a^{19} + \frac{851}{2069} a^{17} - \frac{562}{2069} a^{16} - \frac{861}{2069} a^{15} + \frac{984}{2069} a^{14} + \frac{505}{2069} a^{13} - \frac{316}{2069} a^{12} - \frac{989}{2069} a^{11} + \frac{3}{2069} a^{10} + \frac{261}{2069} a^{9} - \frac{582}{2069} a^{8} + \frac{831}{2069} a^{7} - \frac{753}{2069} a^{6} + \frac{867}{2069} a^{5} - \frac{438}{2069} a^{4} + \frac{123}{2069} a^{3} + \frac{113}{2069} a^{2} + \frac{723}{2069} a + \frac{479}{2069}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{38788}{2069} a^{19} - \frac{148680}{2069} a^{18} + \frac{402748}{2069} a^{17} - \frac{910972}{2069} a^{16} + \frac{1769648}{2069} a^{15} - \frac{2936972}{2069} a^{14} + \frac{4228104}{2069} a^{13} - \frac{5212568}{2069} a^{12} + \frac{5437540}{2069} a^{11} - \frac{4654497}{2069} a^{10} + \frac{3243524}{2069} a^{9} - \frac{560548}{2069} a^{8} - \frac{1310568}{2069} a^{7} + \frac{1419928}{2069} a^{6} - \frac{683128}{2069} a^{5} + \frac{20216}{2069} a^{4} + \frac{326704}{2069} a^{3} - \frac{332920}{2069} a^{2} + \frac{163336}{2069} a - \frac{38572}{2069} \) (order $22$)
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:  \( 6060.61866235 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times D_4$ (as 20T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 25 conjugacy class representatives for $C_5\times D_4$
Character table for $C_5\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.605.1, \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\)

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

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed