# Properties

 Label 43681j Number of curves $3$ Conductor $43681$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 43681j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
43681.i3 43681j1 $$[0, -1, 1, 29121, -94238]$$ $$32768/19$$ $$-1583548311814579$$ $$[]$$ $$162000$$ $$1.6060$$ $$\Gamma_0(N)$$-optimal
43681.i2 43681j2 $$[0, -1, 1, -407689, -106457473]$$ $$-89915392/6859$$ $$-571660940565063019$$ $$[]$$ $$486000$$ $$2.1553$$
43681.i1 43681j3 $$[0, -1, 1, -33605249, -74971104968]$$ $$-50357871050752/19$$ $$-1583548311814579$$ $$[]$$ $$1458000$$ $$2.7046$$

## Rank

sage: E.rank()

The elliptic curves in class 43681j have rank $$1$$.

## Complex multiplication

The elliptic curves in class 43681j do not have complex multiplication.

## Modular form 43681.2.a.j

sage: E.q_eigenform(10)

$$q + 2q^{3} - 2q^{4} + 3q^{5} + q^{7} + q^{9} - 4q^{12} - 4q^{13} + 6q^{15} + 4q^{16} + 3q^{17} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 