# Properties

 Label 786.d Number of curves 4 Conductor 786 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("786.d1")

sage: E.isogeny_class()

## Elliptic curves in class 786.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
786.d1 786e3 [1, 1, 0, -5589, 158517]  576
786.d2 786e2 [1, 1, 0, -349, 2365] [2, 2] 288
786.d3 786e4 [1, 1, 0, -229, 4165]  576
786.d4 786e1 [1, 1, 0, -29, -3]  144 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form786.2.a.d

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 