# Properties

 Label 414736by Number of curves $2$ Conductor $414736$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("by1")

sage: E.isogeny_class()

## Elliptic curves in class 414736by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.by2 414736by1 $$[0, -1, 0, -734428, -311051040]$$ $$-9826000/3703$$ $$-16510070720325269248$$ $$$$ $$8110080$$ $$2.3974$$ $$\Gamma_0(N)$$-optimal*
414736.by1 414736by2 $$[0, -1, 0, -12658088, -17328498592]$$ $$12576878500/1127$$ $$20099216529091632128$$ $$$$ $$16220160$$ $$2.7439$$ $$\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 414736by1.

## Rank

sage: E.rank()

The elliptic curves in class 414736by have rank $$1$$.

## Complex multiplication

The elliptic curves in class 414736by do not have complex multiplication.

## Modular form 414736.2.a.by

sage: E.q_eigenform(10)

$$q + 2q^{3} + q^{9} + 4q^{11} - 6q^{13} + O(q^{20})$$ ## 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. 