# Properties

 Label 41616ca Number of curves 4 Conductor 41616 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("41616.be1")

sage: E.isogeny_class()

## Elliptic curves in class 41616ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
41616.be4 41616ca1 [0, 0, 0, -125715, 10621906]  331776 $$\Gamma_0(N)$$-optimal
41616.be3 41616ca2 [0, 0, 0, -1790355, 921845842]  663552
41616.be2 41616ca3 [0, 0, 0, -4287315, -3416372462]  995328
41616.be1 41616ca4 [0, 0, 0, -4703475, -2713145294]  1990656

## Rank

sage: E.rank()

The elliptic curves in class 41616ca have rank $$1$$.

## Modular form 41616.2.a.be

sage: E.q_eigenform(10)

$$q - 4q^{7} - 6q^{11} + 2q^{13} + 4q^{19} + O(q^{20})$$

## 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 Cremona 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 Cremona labels. 