# Properties

 Label 286650kf Number of curves $6$ Conductor $286650$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("286650.kf1")

sage: E.isogeny_class()

## Elliptic curves in class 286650kf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
286650.kf6 286650kf1 [1, -1, 1, 165145, 10303647] [2] 4718592 $$\Gamma_0(N)$$-optimal
286650.kf5 286650kf2 [1, -1, 1, -716855, 86155647] [2, 2] 9437184
286650.kf2 286650kf3 [1, -1, 1, -9316355, 10938724647] [2] 18874368
286650.kf3 286650kf4 [1, -1, 1, -6229355, -5922469353] [2, 2] 18874368
286650.kf4 286650kf5 [1, -1, 1, -1268105, -15100781853] [2] 37748736
286650.kf1 286650kf6 [1, -1, 1, -99390605, -381362306853] [2] 37748736

## Rank

sage: E.rank()

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

## Modular form 286650.2.a.kf

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{8} - 4q^{11} + q^{13} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.