# Properties

 Label 435344.m Number of curves $2$ Conductor $435344$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 435344.m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.m1 435344m2 $$[0, 1, 0, -37912, 2374932]$$ $$304821217/51842$$ $$1024947946201088$$ $$$$ $$2488320$$ $$1.6009$$ $$\Gamma_0(N)$$-optimal*
435344.m2 435344m1 $$[0, 1, 0, -10872, -404780]$$ $$7189057/644$$ $$12732272623616$$ $$$$ $$1244160$$ $$1.2544$$ $$\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 435344.m1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 435344.2.a.m

sage: E.q_eigenform(10)

$$q - 2q^{3} + 2q^{5} + q^{7} + q^{9} + 6q^{11} - 4q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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 LMFDB 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 LMFDB labels. 