# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("409101.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 409101bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.bd1 409101bd1 [0, 1, 1, -1798463, -929107294] [] 5806080 $$\Gamma_0(N)$$-optimal*
409101.bd2 409101bd2 [0, 1, 1, 691717, -3232399285] [] 17418240 $$\Gamma_0(N)$$-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 409101bd1.

## Rank

sage: E.rank()

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

## Modular form 409101.2.a.bd

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 Cremona 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 Cremona labels.
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