Properties

 Label 439569.be Number of curves $4$ Conductor $439569$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("be1")

sage: E.isogeny_class()

Elliptic curves in class 439569.be

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
439569.be1 439569be3 $$[1, -1, 1, -39863414, 96884531020]$$ $$82483294977/17$$ $$1443876645515769153$$ $$[2]$$ $$21233664$$ $$2.8717$$ $$\Gamma_0(N)$$-optimal*
439569.be2 439569be2 $$[1, -1, 1, -2500049, 1503332848]$$ $$20346417/289$$ $$24545902973768075601$$ $$[2, 2]$$ $$10616832$$ $$2.5252$$ $$\Gamma_0(N)$$-optimal*
439569.be3 439569be1 $$[1, -1, 1, -302204, -27246410]$$ $$35937/17$$ $$1443876645515769153$$ $$[2]$$ $$5308416$$ $$2.1786$$ $$\Gamma_0(N)$$-optimal*
439569.be4 439569be4 $$[1, -1, 1, -302204, 4052833048]$$ $$-35937/83521$$ $$-7093765959418973848689$$ $$[2]$$ $$21233664$$ $$2.8717$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 439569.be1.

Rank

sage: E.rank()

The elliptic curves in class 439569.be have rank $$0$$.

Complex multiplication

The elliptic curves in class 439569.be do not have complex multiplication.

Modular form 439569.2.a.be

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + 3q^{8} - 2q^{10} - 4q^{14} - q^{16} + 4q^{19} + O(q^{20})$$

Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.