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Properties

Label 20.0.19706833839...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 71^{8}$
Root discriminant $18.40$
Ramified primes $5, 71$
Class number $1$
Class group Trivial
Galois group $C_4\times A_5$ (as 20T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 13, -23, 49, -74, 91, -107, 133, -138, 111, -84, 79, -58, 32, -14, 14, -17, 12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 12*x^18 - 17*x^17 + 14*x^16 - 14*x^15 + 32*x^14 - 58*x^13 + 79*x^12 - 84*x^11 + 111*x^10 - 138*x^9 + 133*x^8 - 107*x^7 + 91*x^6 - 74*x^5 + 49*x^4 - 23*x^3 + 13*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 12*x^18 - 17*x^17 + 14*x^16 - 14*x^15 + 32*x^14 - 58*x^13 + 79*x^12 - 84*x^11 + 111*x^10 - 138*x^9 + 133*x^8 - 107*x^7 + 91*x^6 - 74*x^5 + 49*x^4 - 23*x^3 + 13*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 12 x^{18} - 17 x^{17} + 14 x^{16} - 14 x^{15} + 32 x^{14} - 58 x^{13} + 79 x^{12} - 84 x^{11} + 111 x^{10} - 138 x^{9} + 133 x^{8} - 107 x^{7} + 91 x^{6} - 74 x^{5} + 49 x^{4} - 23 x^{3} + 13 x^{2} - 5 x + 1 \)

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:  \(19706833839287139892578125=5^{15}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.40$
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:  $5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{2}{11} a^{17} + \frac{3}{11} a^{16} + \frac{2}{11} a^{15} + \frac{3}{11} a^{14} + \frac{5}{11} a^{13} - \frac{2}{11} a^{12} + \frac{4}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} + \frac{2}{11} a^{7} + \frac{3}{11} a^{6} + \frac{3}{11} a^{3} + \frac{4}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{67283756419} a^{19} + \frac{126764977}{6116705129} a^{18} + \frac{27527087444}{67283756419} a^{17} - \frac{23513897587}{67283756419} a^{16} + \frac{29322294209}{67283756419} a^{15} + \frac{17924959799}{67283756419} a^{14} + \frac{9543452566}{67283756419} a^{13} - \frac{26215314088}{67283756419} a^{12} - \frac{9046675777}{67283756419} a^{11} - \frac{25803157037}{67283756419} a^{10} + \frac{17303214437}{67283756419} a^{9} + \frac{740827971}{6116705129} a^{8} + \frac{27732285118}{67283756419} a^{7} + \frac{6775375447}{67283756419} a^{6} + \frac{535692736}{6116705129} a^{5} + \frac{1769665230}{67283756419} a^{4} - \frac{6401084780}{67283756419} a^{3} - \frac{25161121431}{67283756419} a^{2} + \frac{593969527}{67283756419} a - \frac{19101714436}{67283756419}$

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:  \( -\frac{52362228436}{67283756419} a^{19} + \frac{232281296431}{67283756419} a^{18} - \frac{478663578321}{67283756419} a^{17} + \frac{525288656151}{67283756419} a^{16} - \frac{209902058657}{67283756419} a^{15} + \frac{304597175996}{67283756419} a^{14} - \frac{1287215522816}{67283756419} a^{13} + \frac{2154183911123}{67283756419} a^{12} - \frac{2452796374615}{67283756419} a^{11} + \frac{2033911366495}{67283756419} a^{10} - \frac{298601207809}{6116705129} a^{9} + \frac{3987737430245}{67283756419} a^{8} - \frac{3062865177124}{67283756419} a^{7} + \frac{1886855302890}{67283756419} a^{6} - \frac{176807117828}{6116705129} a^{5} + \frac{1563593580370}{67283756419} a^{4} - \frac{811223099221}{67283756419} a^{3} + \frac{169793817286}{67283756419} a^{2} - \frac{273722947127}{67283756419} a + \frac{58027725656}{67283756419} \) (order $10$)
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:  \( 183267.118192 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times A_5$ (as 20T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 20 conjugacy class representatives for $C_4\times A_5$
Character table for $C_4\times A_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.126025.1, 10.2.79411503125.1

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

Sibling fields

Degree 24 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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$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$$[\ ]_{4}^{2}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.3.2.1$x^{3} - 71$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
71.3.2.1$x^{3} - 71$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
71.3.2.1$x^{3} - 71$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
71.3.2.1$x^{3} - 71$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$