# Properties

 Label 165649.f Number of curves $2$ Conductor $165649$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 165649.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
165649.f1 165649f2 [1, 1, 1, -4972921, 4266480472] [] 3409560
165649.f2 165649f1 [1, 1, 1, -3451, -306470] [] 309960 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 165649.f have rank $$0$$.

## Complex multiplication

The elliptic curves in class 165649.f do not have complex multiplication.

## Modular form 165649.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + 2q^{3} - q^{4} - q^{5} - 2q^{6} - 2q^{7} + 3q^{8} + q^{9} + q^{10} - 2q^{12} - q^{13} + 2q^{14} - 2q^{15} - q^{16} + 5q^{17} - q^{18} - 6q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 