# Properties

 Label 409101s Number of curves $2$ Conductor $409101$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("409101.s1")

sage: E.isogeny_class()

## Elliptic curves in class 409101s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.s2 409101s1 [1, 0, 0, -429976, -171169657] [2] 12533760 $$\Gamma_0(N)$$-optimal*
409101.s1 409101s2 [1, 0, 0, -7930161, -8593877412] [2] 25067520 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 409101s1.

## Rank

sage: E.rank()

The elliptic curves in class 409101s have rank $$0$$.

## Modular form 409101.2.a.s

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.