# Properties

 Label 66654w Number of curves $6$ Conductor $66654$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("66654.j1")

sage: E.isogeny_class()

## Elliptic curves in class 66654w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
66654.j5 66654w1 [1, -1, 0, -19143, 2256061] [2] 360448 $$\Gamma_0(N)$$-optimal
66654.j4 66654w2 [1, -1, 0, -400023, 97399885] [2, 2] 720896
66654.j3 66654w3 [1, -1, 0, -495243, 47599825] [2, 2] 1441792
66654.j1 66654w4 [1, -1, 0, -6398883, 6231834121] [2] 1441792
66654.j6 66654w5 [1, -1, 0, 1837647, 366272599] [2] 2883584
66654.j2 66654w6 [1, -1, 0, -4351653, -3459419429] [2] 2883584

## Rank

sage: E.rank()

The elliptic curves in class 66654w have rank $$0$$.

## Modular form 66654.2.a.j

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 2q^{5} + q^{7} - q^{8} + 2q^{10} - 4q^{11} + 6q^{13} - q^{14} + q^{16} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.