↑

Properties

Label 20.0.87423165143...4281.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 1567^{6}$
Root discriminant $15.74$
Ramified primes $3, 1567$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T288

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 2, 10, 13, 49, 129, -12, 332, -457, 386, -614, 371, -63, 136, -136, 34, -10, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 15*x^18 - 10*x^17 + 34*x^16 - 136*x^15 + 136*x^14 - 63*x^13 + 371*x^12 - 614*x^11 + 386*x^10 - 457*x^9 + 332*x^8 - 12*x^7 + 129*x^6 + 49*x^5 + 13*x^4 + 10*x^3 + 2*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 15*x^18 - 10*x^17 + 34*x^16 - 136*x^15 + 136*x^14 - 63*x^13 + 371*x^12 - 614*x^11 + 386*x^10 - 457*x^9 + 332*x^8 - 12*x^7 + 129*x^6 + 49*x^5 + 13*x^4 + 10*x^3 + 2*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 15 x^{18} - 10 x^{17} + 34 x^{16} - 136 x^{15} + 136 x^{14} - 63 x^{13} + 371 x^{12} - 614 x^{11} + 386 x^{10} - 457 x^{9} + 332 x^{8} - 12 x^{7} + 129 x^{6} + 49 x^{5} + 13 x^{4} + 10 x^{3} + 2 x^{2} + x + 1 \)

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:  \(874231651432481601634281=3^{10}\cdot 1567^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.74$
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:  $3, 1567$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{148731819873824917} a^{19} - \frac{40747443773618292}{148731819873824917} a^{18} - \frac{14998647865014271}{148731819873824917} a^{17} - \frac{64436988471893294}{148731819873824917} a^{16} + \frac{18084632589197150}{148731819873824917} a^{15} - \frac{33888769116699080}{148731819873824917} a^{14} - \frac{14520046567734238}{148731819873824917} a^{13} - \frac{5813718783277617}{148731819873824917} a^{12} - \frac{34031415124392144}{148731819873824917} a^{11} + \frac{58771864471994980}{148731819873824917} a^{10} + \frac{35129207982910712}{148731819873824917} a^{9} + \frac{54867219453810918}{148731819873824917} a^{8} - \frac{54474909445098326}{148731819873824917} a^{7} - \frac{6177765967072276}{148731819873824917} a^{6} + \frac{47726560638371105}{148731819873824917} a^{5} + \frac{26820099473390109}{148731819873824917} a^{4} - \frac{21863988723107195}{148731819873824917} a^{3} - \frac{8628116740975497}{148731819873824917} a^{2} - \frac{44570765784835613}{148731819873824917} a + \frac{60942846875657191}{148731819873824917}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{6043180895}{11914651339} a^{19} - \frac{44482799536}{11914651339} a^{18} + \frac{104346213914}{11914651339} a^{17} - \frac{80375439309}{11914651339} a^{16} + \frac{191118616181}{11914651339} a^{15} - \frac{853886919079}{11914651339} a^{14} + \frac{1052489505011}{11914651339} a^{13} - \frac{415203197655}{11914651339} a^{12} + \frac{1928007199287}{11914651339} a^{11} - \frac{4196239217689}{11914651339} a^{10} + \frac{3081171388921}{11914651339} a^{9} - \frac{2086582787223}{11914651339} a^{8} + \frac{1316987436521}{11914651339} a^{7} + \frac{381223632518}{11914651339} a^{6} - \frac{46574356203}{11914651339} a^{5} + \frac{320153896977}{11914651339} a^{4} - \frac{42366794492}{11914651339} a^{3} + \frac{41178034543}{11914651339} a^{2} + \frac{28558669325}{11914651339} a + \frac{4068674534}{11914651339} \) (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:  \( 11086.8488736 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.3.14103.1, 10.0.596683827.1, 10.2.935003556909.1, 10.4.311667852303.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1567Data not computed