# Properties

 Label 17661.f Number of curves 6 Conductor 17661 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("17661.f1")

sage: E.isogeny_class()

## Elliptic curves in class 17661.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
17661.f1 17661a5 [1, 1, 0, -659361, -206353626] [2] 100352
17661.f2 17661a3 [1, 1, 0, -41226, -3234465] [2, 2] 50176
17661.f3 17661a4 [1, 1, 0, -32816, 2260629] [2] 50176
17661.f4 17661a6 [1, 1, 0, -28611, -5235204] [2] 100352
17661.f5 17661a2 [1, 1, 0, -3381, -17640] [2, 2] 25088
17661.f6 17661a1 [1, 1, 0, 824, -1661] [2] 12544 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 17661.f have rank $$1$$.

## Modular form 17661.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} - 3q^{8} + q^{9} - 2q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{14} + 2q^{15} - q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.