# Properties

 Label 317680.s Number of curves 4 Conductor 317680 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("317680.s1")

sage: E.isogeny_class()

## Elliptic curves in class 317680.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
317680.s1 317680s4 [0, 1, 0, -2563220, -1580381000]  5971968
317680.s2 317680s3 [0, 1, 0, -160765, -24551142]  2985984
317680.s3 317680s2 [0, 1, 0, -36220, -1511400]  1990656
317680.s4 317680s1 [0, 1, 0, -16365, 783838]  995328 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 317680.s have rank $$1$$.

## Modular form 317680.2.a.s

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{5} + 4q^{7} + q^{9} + q^{11} + 4q^{13} - 2q^{15} + O(q^{20})$$

## 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. 