# Properties

 Label 409101.y Number of curves $2$ Conductor $409101$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("409101.y1")

sage: E.isogeny_class()

## Elliptic curves in class 409101.y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.y1 409101y1 [0, -1, 1, -36703, 2719254] [] 829440 $$\Gamma_0(N)$$-optimal
409101.y2 409101y2 [0, -1, 1, 14117, 9419871] [] 2488320

## Rank

sage: E.rank()

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

## Modular form 409101.2.a.y

sage: E.q_eigenform(10)

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

## 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. 