# Properties

 Label 41650.f Number of curves 4 Conductor 41650 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("41650.f1")

sage: E.isogeny_class()

## Elliptic curves in class 41650.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
41650.f1 41650n4 [1, 0, 1, -138451, 13690548]  497664
41650.f2 41650n3 [1, 0, 1, -126201, 17243048]  248832
41650.f3 41650n2 [1, 0, 1, -52701, -4659952]  165888
41650.f4 41650n1 [1, 0, 1, -3701, -53952]  82944 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 41650.f have rank $$0$$.

## Modular form 41650.2.a.f

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 