# Properties

 Label 286650k Number of curves 4 Conductor 286650 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("286650.k1")

sage: E.isogeny_class()

## Elliptic curves in class 286650k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
286650.k4 286650k1 [1, -1, 0, -4301817, -2894416659]  15925248 $$\Gamma_0(N)$$-optimal
286650.k3 286650k2 [1, -1, 0, -19589817, 30632167341]  31850496
286650.k2 286650k3 [1, -1, 0, -96911817, 366924753341]  47775744
286650.k1 286650k4 [1, -1, 0, -1550227317, 23493534304841]  95551488

## Rank

sage: E.rank()

The elliptic curves in class 286650k have rank $$1$$.

## Modular form 286650.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{8} - 6q^{11} + q^{13} + q^{16} + 4q^{19} + O(q^{20})$$

## 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. 