# Properties

 Label 435344bd Number of curves $4$ Conductor $435344$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bd1")

sage: E.isogeny_class()

## Elliptic curves in class 435344bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.bd4 435344bd1 $$[0, 0, 0, -24674, -2603445]$$ $$-21511084032/25465531$$ $$-1966676067529264$$ $$$$ $$1376256$$ $$1.6285$$ $$\Gamma_0(N)$$-optimal*
435344.bd3 435344bd2 $$[0, 0, 0, -471679, -124635810]$$ $$9392111857872/4380649$$ $$5413006340874496$$ $$[2, 2]$$ $$2752512$$ $$1.9751$$ $$\Gamma_0(N)$$-optimal*
435344.bd2 435344bd3 $$[0, 0, 0, -549419, -80774902]$$ $$3710860803108/1577224103$$ $$7795670523266382848$$ $$$$ $$5505024$$ $$2.3217$$ $$\Gamma_0(N)$$-optimal*
435344.bd1 435344bd4 $$[0, 0, 0, -7546019, -7978568078]$$ $$9614292367656708/2093$$ $$10344971506688$$ $$$$ $$5505024$$ $$2.3217$$
*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 435344bd1.

## Rank

sage: E.rank()

The elliptic curves in class 435344bd have rank $$1$$.

## Complex multiplication

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

## Modular form 435344.2.a.bd

sage: E.q_eigenform(10)

$$q + 2q^{5} - q^{7} - 3q^{9} + 4q^{11} - 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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 Cremona 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 Cremona labels. 