# Properties

 Label 850.e Number of curves 4 Conductor 850 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("850.e1")

sage: E.isogeny_class()

## Elliptic curves in class 850.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
850.e1 850b4 [1, 1, 0, -2825, -41125]  1728
850.e2 850b3 [1, 1, 0, -2575, -51375]  864
850.e3 850b2 [1, 1, 0, -1075, 13125]  576
850.e4 850b1 [1, 1, 0, -75, 125]  288 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 850.e have rank $$0$$.

## Modular form850.2.a.e

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 