# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("v1")

sage: E.isogeny_class()

## Elliptic curves in class 435344v

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.v1 435344v1 $$[0, -1, 0, -118130549, -494147779139]$$ $$-9221261135586623488/121324931$$ $$-2398667853312733184$$ $$[]$$ $$41803776$$ $$3.0852$$ $$\Gamma_0(N)$$-optimal*
435344.v2 435344v2 $$[0, -1, 0, -111451669, -552492436003]$$ $$-7743965038771437568/2189290237869371$$ $$-43283602734121045062201344$$ $$[]$$ $$125411328$$ $$3.6345$$ $$\Gamma_0(N)$$-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 435344v1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 435344.2.a.v

sage: E.q_eigenform(10)

$$q - q^{3} + 3q^{5} + q^{7} - 2q^{9} + 3q^{11} - 3q^{15} + 6q^{17} - q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 