# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("786.k1")

sage: E.isogeny_class()

## Elliptic curves in class 786.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
786.k1 786i1 [1, 1, 1, -71, -259]  288 $$\Gamma_0(N)$$-optimal
786.k2 786i2 [1, 1, 1, -31, -499]  576

## Rank

sage: E.rank()

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

## Modular form786.2.a.k

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 