# Properties

 Label 50.a Number of curves 4 Conductor 50 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("50.a1")

sage: E.isogeny_class()

## Elliptic curves in class 50.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
50.a1 50a2 [1, 0, 1, -126, -552] [] 6
50.a2 50a3 [1, 0, 1, -76, 298]  10
50.a3 50a1 [1, 0, 1, -1, -2]  2 $$\Gamma_0(N)$$-optimal
50.a4 50a4 [1, 0, 1, 549, -2202] [] 30

## Rank

sage: E.rank()

The elliptic curves in class 50.a have rank $$0$$.

## Modular form50.2.a.a

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 