# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("17661.h1")

sage: E.isogeny_class()

## Elliptic curves in class 17661f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
17661.h5 17661f1 [1, 0, 1, -659362, 220588751] [2] 322560 $$\Gamma_0(N)$$-optimal
17661.h4 17661f2 [1, 0, 1, -10755567, 13575848725] [2, 2] 645120
17661.h3 17661f3 [1, 0, 1, -10961612, 13028593205] [2, 2] 1290240
17661.h1 17661f4 [1, 0, 1, -172088802, 868900127401] [2] 1290240
17661.h2 17661f5 [1, 0, 1, -35716447, -66790896769] [2] 2580480
17661.h6 17661f6 [1, 0, 1, 10496503, 57824554079] [2] 2580480

## Rank

sage: E.rank()

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

## Modular form 17661.2.a.h

sage: E.q_eigenform(10)

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

## 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.