↑

Properties

Label 20.12.4557142893...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{4}\cdot 9931^{4}$
Root discriminant $68.07$
Ramified primes $2, 5, 11, 9931$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1010

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 0, 125, 0, -30200, 0, 67725, 0, -34485, 0, -3580, 0, 4539, 0, -139, 0, -125, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 125*x^16 - 139*x^14 + 4539*x^12 - 3580*x^10 - 34485*x^8 + 67725*x^6 - 30200*x^4 + 125*x^2 + 125)
 
gp: K = bnfinit(x^20 + 3*x^18 - 125*x^16 - 139*x^14 + 4539*x^12 - 3580*x^10 - 34485*x^8 + 67725*x^6 - 30200*x^4 + 125*x^2 + 125, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} - 125 x^{16} - 139 x^{14} + 4539 x^{12} - 3580 x^{10} - 34485 x^{8} + 67725 x^{6} - 30200 x^{4} + 125 x^{2} + 125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4557142893582673873952000000000000000=2^{20}\cdot 5^{15}\cdot 11^{4}\cdot 9931^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.07$
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:  $2, 5, 11, 9931$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4}$, $\frac{1}{25} a^{15} - \frac{2}{25} a^{13} - \frac{1}{5} a^{11} - \frac{9}{25} a^{9} + \frac{9}{25} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{16} - \frac{2}{25} a^{14} + \frac{6}{25} a^{10} + \frac{9}{25} a^{8} - \frac{2}{5} a^{6}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{13} + \frac{11}{25} a^{11} - \frac{9}{25} a^{9} - \frac{12}{25} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3}$, $\frac{1}{2350504462659437125} a^{18} + \frac{17645375470499453}{2350504462659437125} a^{16} - \frac{34169977313424989}{470100892531887425} a^{14} + \frac{40631604184064601}{2350504462659437125} a^{12} - \frac{186275925195785486}{2350504462659437125} a^{10} - \frac{2880781955011113}{94020178506377485} a^{8} - \frac{106950872501146523}{470100892531887425} a^{6} + \frac{26262540892903124}{94020178506377485} a^{4} + \frac{41335424548988519}{94020178506377485} a^{2} - \frac{1105721805973734}{18804035701275497}$, $\frac{1}{2350504462659437125} a^{19} + \frac{17645375470499453}{2350504462659437125} a^{17} + \frac{687618817825201}{94020178506377485} a^{15} + \frac{134651782690442086}{2350504462659437125} a^{13} + \frac{283824967336101939}{2350504462659437125} a^{11} + \frac{117224340133872914}{470100892531887425} a^{9} - \frac{144558943903697517}{470100892531887425} a^{7} - \frac{2269106101929574}{18804035701275497} a^{5} - \frac{15076682554837972}{94020178506377485} a^{3} - \frac{1105721805973734}{18804035701275497} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 182246339595 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

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 $20$ R $20$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
5Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
9931Data not computed