# Properties

 Label 65.a Number of curves $2$ Conductor $65$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

## Elliptic curves in class 65.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
65.a1 65a1 $$[1, 0, 0, -1, 0]$$ $$117649/65$$ $$65$$ $$$$ $$2$$ $$-0.96162$$ $$\Gamma_0(N)$$-optimal
65.a2 65a2 $$[1, 0, 0, 4, 1]$$ $$6967871/4225$$ $$-4225$$ $$$$ $$4$$ $$-0.61504$$

## Rank

sage: E.rank()

The elliptic curves in class 65.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 65.a do not have complex multiplication.

## Modular form65.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 2 q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{10} + 2 q^{11} + 2 q^{12} - q^{13} + 4 q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} - 6 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 