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Properties

Label 20.0.21863652182...4881.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 1483^{2}\cdot 129751961^{2}$
Root discriminant $23.28$
Ramified primes $3, 1483, 129751961$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 10, -5, 58, -41, 200, -216, 465, -427, 571, -547, 496, -369, 258, -146, 80, -34, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 15*x^18 - 34*x^17 + 80*x^16 - 146*x^15 + 258*x^14 - 369*x^13 + 496*x^12 - 547*x^11 + 571*x^10 - 427*x^9 + 465*x^8 - 216*x^7 + 200*x^6 - 41*x^5 + 58*x^4 - 5*x^3 + 10*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 15*x^18 - 34*x^17 + 80*x^16 - 146*x^15 + 258*x^14 - 369*x^13 + 496*x^12 - 547*x^11 + 571*x^10 - 427*x^9 + 465*x^8 - 216*x^7 + 200*x^6 - 41*x^5 + 58*x^4 - 5*x^3 + 10*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 15 x^{18} - 34 x^{17} + 80 x^{16} - 146 x^{15} + 258 x^{14} - 369 x^{13} + 496 x^{12} - 547 x^{11} + 571 x^{10} - 427 x^{9} + 465 x^{8} - 216 x^{7} + 200 x^{6} - 41 x^{5} + 58 x^{4} - 5 x^{3} + 10 x^{2} + x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2186365218234941887328764881=3^{10}\cdot 1483^{2}\cdot 129751961^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.28$
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:  $3, 1483, 129751961$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{21846140761656067} a^{19} + \frac{6396475296551841}{21846140761656067} a^{18} + \frac{5756924221204748}{21846140761656067} a^{17} + \frac{8836129689108227}{21846140761656067} a^{16} + \frac{7991570044550675}{21846140761656067} a^{15} - \frac{2845744487615082}{21846140761656067} a^{14} + \frac{6825371631016091}{21846140761656067} a^{13} - \frac{2985108977632982}{21846140761656067} a^{12} - \frac{7942656265500918}{21846140761656067} a^{11} - \frac{8225173701390150}{21846140761656067} a^{10} + \frac{9935733163163954}{21846140761656067} a^{9} + \frac{7613793304802984}{21846140761656067} a^{8} + \frac{7022371768991133}{21846140761656067} a^{7} + \frac{8229495676851394}{21846140761656067} a^{6} - \frac{3051114769321261}{21846140761656067} a^{5} - \frac{5553223210360596}{21846140761656067} a^{4} + \frac{1088513024704683}{21846140761656067} a^{3} - \frac{8645550356210408}{21846140761656067} a^{2} + \frac{2953632597863099}{21846140761656067} a + \frac{3300439422227443}{21846140761656067}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{5721412821848745}{21846140761656067} a^{19} - \frac{25393623093033495}{21846140761656067} a^{18} + \frac{94729798118001009}{21846140761656067} a^{17} - \frac{227360959046271943}{21846140761656067} a^{16} + \frac{525269398347439700}{21846140761656067} a^{15} - \frac{994799518034214901}{21846140761656067} a^{14} + \frac{1747711591383955892}{21846140761656067} a^{13} - \frac{2583493760443542936}{21846140761656067} a^{12} + \frac{3459283672189962487}{21846140761656067} a^{11} - \frac{3937515771798912192}{21846140761656067} a^{10} + \frac{4066803286216729302}{21846140761656067} a^{9} - \frac{3251825415619390177}{21846140761656067} a^{8} + \frac{3107005929357761800}{21846140761656067} a^{7} - \frac{1951400815693389506}{21846140761656067} a^{6} + \frac{1231902654220950053}{21846140761656067} a^{5} - \frac{504901316536654208}{21846140761656067} a^{4} + \frac{282455009594543940}{21846140761656067} a^{3} - \frac{166288748869849943}{21846140761656067} a^{2} + \frac{36584548568516604}{21846140761656067} a - \frac{2031486049070975}{21846140761656067} \) (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:  \( 244847.728749 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.8.192422158163.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $18{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ $18{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
1483Data not computed
129751961Data not computed