# Properties

 Label 435344e Number of curves $2$ Conductor $435344$ CM no Rank $2$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 435344e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.e2 435344e1 $$[0, 1, 0, -444864, -114354044]$$ $$1969910093092/7889$$ $$38992584909824$$ $$$$ $$2073600$$ $$1.8193$$ $$\Gamma_0(N)$$-optimal*
435344.e1 435344e2 $$[0, 1, 0, -451624, -110706348]$$ $$1030541881826/62236321$$ $$615225004707203072$$ $$$$ $$4147200$$ $$2.1658$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 435344e1.

## Rank

sage: E.rank()

The elliptic curves in class 435344e have rank $$2$$.

## Complex multiplication

The elliptic curves in class 435344e do not have complex multiplication.

## Modular form 435344.2.a.e

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{5} + q^{7} + q^{9} + 2q^{11} + 4q^{15} - 2q^{17} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 