# Properties

 Label 435344x Number of curves $2$ Conductor $435344$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 435344x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.x2 435344x1 $$[0, 0, 0, -9971, -435006]$$ $$-22180932/3703$$ $$-18302641896448$$ $$$$ $$552960$$ $$1.2718$$ $$\Gamma_0(N)$$-optimal*
435344.x1 435344x2 $$[0, 0, 0, -165451, -25902630]$$ $$50668941906/1127$$ $$11140738545664$$ $$$$ $$1105920$$ $$1.6184$$ $$\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 435344x1.

## Rank

sage: E.rank()

The elliptic curves in class 435344x have rank $$0$$.

## Complex multiplication

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

## Modular form 435344.2.a.x

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 