# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("409101.bt1")

sage: E.isogeny_class()

## Elliptic curves in class 409101.bt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.bt1 409101bt2 [1, 0, 1, -7340226, -7595485691]  13824000 $$\Gamma_0(N)$$-optimal*
409101.bt2 409101bt1 [1, 0, 1, -136491, -282253919]  6912000 $$\Gamma_0(N)$$-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.bt2.

## Rank

sage: E.rank()

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

## Modular form 409101.2.a.bt

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 