Properties

 Label 86190.e2 Conductor 86190 Discriminant -2571478650 j-invariant $$-\frac{16194277}{1170450}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -68, -2478]); // or

magma: E := EllipticCurve("86190j2");

sage: E = EllipticCurve([1, 1, 0, -68, -2478]) # or

sage: E = EllipticCurve("86190j2")

gp: E = ellinit([1, 1, 0, -68, -2478]) \\ or

gp: E = ellinit("86190j2")

$$y^2 + x y = x^{3} + x^{2} - 68 x - 2478$$

Mordell-Weil group structure

$$\Z^2 \times \Z/{2}\Z$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(17, 32\right)$$ $$\left(19, 50\right)$$ $$\hat{h}(P)$$ ≈ 2.20055499504 1.95139767793

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(\frac{59}{4}, -\frac{59}{8}\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(17, 32\right)$$, $$\left(17, -49\right)$$, $$\left(19, 50\right)$$, $$\left(19, -69\right)$$, $$\left(31, 147\right)$$, $$\left(31, -178\right)$$, $$\left(53, 356\right)$$, $$\left(53, -409\right)$$, $$\left(161, 1967\right)$$, $$\left(161, -2128\right)$$, $$\left(291, 4827\right)$$, $$\left(291, -5118\right)$$

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: $$-2571478650$$ = $$-1 \cdot 2 \cdot 3^{4} \cdot 5^{2} \cdot 13^{3} \cdot 17^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{16194277}{1170450}$$ = $$-1 \cdot 2^{-1} \cdot 3^{-4} \cdot 5^{-2} \cdot 11^{3} \cdot 17^{-2} \cdot 23^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$2.69160083179$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.636240780976$$ 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: $$16$$  = $$1\cdot2\cdot2\cdot2\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 86190.2.a.e

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 49152 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$6.85002486118$$

Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$13$$ $$2$$ $$III$$ Additive -1 2 3 0
$$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

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 X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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
$$2$$ B

$p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 nonsplit nonsplit nonsplit ss ss add nonsplit ordinary ss ordinary ordinary ordinary ordinary ordinary ss 4 2 2 2,2 2,2 - 2 2 2,2 2 2 2 2 2 2,2 1 0 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.e consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-26})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.2.507946400.5 $$\Z/4\Z$$ 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.