# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 435344.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.p1 435344p1 $$[0, -1, 0, -31997, -5187599]$$ $$-2932006912/7750379$$ $$-9576857372316416$$ $$[]$$ $$2322432$$ $$1.7541$$ $$\Gamma_0(N)$$-optimal*
435344.p2 435344p2 $$[0, -1, 0, 278963, 117330641]$$ $$1942951190528/5944921619$$ $$-7345920300770245376$$ $$[]$$ $$6967296$$ $$2.3034$$ $$\Gamma_0(N)$$-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.p1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 435344.2.a.p

sage: E.q_eigenform(10)

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