# Properties

 Label 116610y Number of curves $6$ Conductor $116610$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("116610.bb1")

sage: E.isogeny_class()

## Elliptic curves in class 116610y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
116610.bb5 116610y1 [1, 0, 1, -70984, 7980182] [2] 884736 $$\Gamma_0(N)$$-optimal
116610.bb4 116610y2 [1, 0, 1, -1166104, 484576406] [2, 2] 1769472
116610.bb3 116610y3 [1, 0, 1, -1196524, 457952822] [2, 2] 3538944
116610.bb1 116610y4 [1, 0, 1, -18657604, 31017738806] [2] 3538944
116610.bb6 116610y5 [1, 0, 1, 1485506, 2219510126] [2] 7077888
116610.bb2 116610y6 [1, 0, 1, -4365274, -3007392178] [2] 7077888

## Rank

sage: E.rank()

The elliptic curves in class 116610y have rank $$2$$.

## Modular form 116610.2.a.bb

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.