# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("50.b1")

sage: E.isogeny_class()

## Elliptic curves in class 50b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
50.b3 50b1 [1, 1, 1, -3, 1]  2 $$\Gamma_0(N)$$-optimal
50.b4 50b2 [1, 1, 1, 22, -9]  6
50.b2 50b3 [1, 1, 1, -13, -219] [] 10
50.b1 50b4 [1, 1, 1, -3138, -68969] [] 30

## Rank

sage: E.rank()

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

## Modular form50.2.a.b

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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 