# Properties

 Label 243568.s Number of curves 3 Conductor 243568 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("243568.s1")

sage: E.isogeny_class()

## Elliptic curves in class 243568.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
243568.s1 243568s3 [0, -1, 0, -60721128, 189144181616] [] 24074496
243568.s2 243568s1 [0, -1, 0, -569608, -186665232] [] 2674944 $$\Gamma_0(N)$$-optimal
243568.s3 243568s2 [0, -1, 0, 3824392, 692697200] [] 8024832

## Rank

sage: E.rank()

The elliptic curves in class 243568.s have rank $$1$$.

## Modular form 243568.2.a.s

sage: E.q_eigenform(10)

$$q + 2q^{3} + q^{7} + q^{9} - 3q^{11} + q^{13} - 3q^{17} - 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 