# Properties

 Label 409101.u Number of curves $2$ Conductor $409101$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("409101.u1")

sage: E.isogeny_class()

## Elliptic curves in class 409101.u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.u1 409101u2 [0, -1, 1, -504968977, 4367604502722] [] 98537472 $$\Gamma_0(N)$$-optimal*
409101.u2 409101u1 [0, -1, 1, -11913337, -6476975583] [] 32845824 $$\Gamma_0(N)$$-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 409101.u2.

## Rank

sage: E.rank()

The elliptic curves in class 409101.u have rank $$1$$.

## Modular form 409101.2.a.u

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.