Properties

Label 86190.b3
Conductor 86190
Discriminant 79467584967934862400
j-invariant \( \frac{29263955267177281}{16463793153600} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -1084983, 72119637]); // or
 
magma: E := EllipticCurve("86190c2");
 
sage: E = EllipticCurve([1, 1, 0, -1084983, 72119637]) # or
 
sage: E = EllipticCurve("86190c2")
 
gp: E = ellinit([1, 1, 0, -1084983, 72119637]) \\ or
 
gp: E = ellinit("86190c2")
 

\( y^2 + x y = x^{3} + x^{2} - 1084983 x + 72119637 \)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
 
sage: E.gens()
 

\(P\) =  \( \left(6, 8097\right) \)
\(\hat{h}(P)\) ≈  3.47839329414

Torsion generators

magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 

\( \left(1006, -503\right) \), \( \left(\frac{267}{4}, -\frac{267}{8}\right) \)

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 

\( \left(-1074, 537\right) \), \( \left(6, 8097\right) \), \( \left(6, -8103\right) \), \( \left(1006, -503\right) \), \( \left(1123, 15916\right) \), \( \left(1123, -17039\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 86190 \)  =  \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(79467584967934862400 \)  =  \(2^{6} \cdot 3^{6} \cdot 5^{2} \cdot 13^{8} \cdot 17^{4} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{29263955267177281}{16463793153600} \)  =  \(2^{-6} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{6} \cdot 13^{-2} \cdot 17^{-4} \cdot 19^{3} \cdot 331^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(3.47839329414\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.166416718146\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 64 \)  = \( 2\cdot2\cdot2\cdot2^{2}\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(4\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 86190.2.a.b

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4q^{7} - q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} + 4q^{14} + q^{15} + q^{16} - q^{17} - q^{18} - 8q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 3870720
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L'(E,1) \) ≈ \( 2.31545118572 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(3\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(5\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(13\) \(4\) \( I_2^{*} \) Additive 1 2 8 2
\(17\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ and has index 6.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit nonsplit nonsplit ordinary ordinary add nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary ss
$\lambda$-invariant(s) 6 7 1 1 3 - 1 1 1 1 1,1 1 1 1 1,1
$\mu$-invariant(s) 0 0 0 0 0 - 0 0 0 0 0,0 0 0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 86190.b consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 \(\Q(\sqrt{10}, \sqrt{13})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-3}, \sqrt{-130})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{3}, \sqrt{-13})\) \(\Z/2\Z \times \Z/4\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.