# Properties

 Label 166410by Number of curves $4$ Conductor $166410$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("by1")

sage: E.isogeny_class()

## Elliptic curves in class 166410by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
166410.a3 166410by1 [1, -1, 0, -1140255, -467780099] [2] 2838528 $$\Gamma_0(N)$$-optimal
166410.a2 166410by2 [1, -1, 0, -1473075, -172169375] [2, 2] 5677056
166410.a1 166410by3 [1, -1, 0, -13953825, 19929326575] [2] 11354112
166410.a4 166410by4 [1, -1, 0, 5682555, -1358572829] [2] 11354112

## Rank

sage: E.rank()

The elliptic curves in class 166410by have rank $$1$$.

## Complex multiplication

The elliptic curves in class 166410by do not have complex multiplication.

## Modular form 166410.2.a.by

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.