↑

Properties

Label 20.2.30065175755...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 3^{6}\cdot 5^{10}\cdot 3319^{5}$
Root discriminant $47.20$
Ramified primes $2, 3, 5, 3319$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2419551, 0, -4903011, 0, -3218373, 0, -1584144, 0, -307944, 0, -46212, 0, -7457, 0, 494, 0, 325, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 33*x^18 + 325*x^16 + 494*x^14 - 7457*x^12 - 46212*x^10 - 307944*x^8 - 1584144*x^6 - 3218373*x^4 - 4903011*x^2 - 2419551)
 
gp: K = bnfinit(x^20 + 33*x^18 + 325*x^16 + 494*x^14 - 7457*x^12 - 46212*x^10 - 307944*x^8 - 1584144*x^6 - 3218373*x^4 - 4903011*x^2 - 2419551, 1)
 

Normalized defining polynomial

\( x^{20} + 33 x^{18} + 325 x^{16} + 494 x^{14} - 7457 x^{12} - 46212 x^{10} - 307944 x^{8} - 1584144 x^{6} - 3218373 x^{4} - 4903011 x^{2} - 2419551 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3006517575591722236631040000000000=-\,2^{20}\cdot 3^{6}\cdot 5^{10}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.20$
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:  $2, 3, 5, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{8} + \frac{2}{9} a^{6} + \frac{1}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{2}{9} a^{7} + \frac{1}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{10} + \frac{2}{27} a^{8} + \frac{10}{27} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{11} + \frac{2}{27} a^{9} + \frac{10}{27} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{567} a^{16} - \frac{2}{189} a^{14} + \frac{19}{567} a^{12} - \frac{31}{567} a^{10} - \frac{65}{567} a^{8} + \frac{10}{63} a^{6} - \frac{4}{63} a^{4} - \frac{10}{21} a^{2} - \frac{2}{7}$, $\frac{1}{567} a^{17} - \frac{2}{189} a^{15} + \frac{19}{567} a^{13} - \frac{31}{567} a^{11} - \frac{65}{567} a^{9} + \frac{10}{63} a^{7} - \frac{4}{63} a^{5} - \frac{10}{21} a^{3} - \frac{2}{7} a$, $\frac{1}{2345528939087211590364907941} a^{18} - \frac{648691074818272672575689}{781842979695737196788302647} a^{16} + \frac{3043767876717303956278021}{335075562726744512909272563} a^{14} - \frac{243978545678060598648802}{335075562726744512909272563} a^{12} + \frac{35443135487517929490669874}{2345528939087211590364907941} a^{10} + \frac{93496545820125614016677569}{260614326565245732262767549} a^{8} + \frac{56385001307678729372957636}{260614326565245732262767549} a^{6} - \frac{9623498742704715985861883}{28957147396138414695863061} a^{4} + \frac{9597926263583906259212960}{28957147396138414695863061} a^{2} - \frac{735830190936652599944302}{9652382465379471565287687}$, $\frac{1}{2345528939087211590364907941} a^{19} - \frac{648691074818272672575689}{781842979695737196788302647} a^{17} + \frac{3043767876717303956278021}{335075562726744512909272563} a^{15} - \frac{243978545678060598648802}{335075562726744512909272563} a^{13} + \frac{35443135487517929490669874}{2345528939087211590364907941} a^{11} + \frac{93496545820125614016677569}{260614326565245732262767549} a^{9} + \frac{56385001307678729372957636}{260614326565245732262767549} a^{7} - \frac{9623498742704715985861883}{28957147396138414695863061} a^{5} + \frac{9597926263583906259212960}{28957147396138414695863061} a^{3} - \frac{735830190936652599944302}{9652382465379471565287687} a$

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:  $10$
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:  \( 206196124.285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.34424253125.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed
3319Data not computed