# Properties

 Label 786.f Number of curves 2 Conductor 786 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("786.f1")

sage: E.isogeny_class()

## Elliptic curves in class 786.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
786.f1 786f2 [1, 0, 1, -145, -580] [] 456
786.f2 786f1 [1, 0, 1, -40, 92]  152 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 786.f have rank $$0$$.

## Modular form786.2.a.f

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 