# Properties

 Label 409101bm Number of curves $6$ Conductor $409101$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("409101.bm1")

sage: E.isogeny_class()

## Elliptic curves in class 409101bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.bm6 409101bm1 [1, 1, 0, 183676, -88756653]  5898240 $$\Gamma_0(N)$$-optimal*
409101.bm5 409101bm2 [1, 1, 0, -2217569, -1146745200] [2, 2] 11796480 $$\Gamma_0(N)$$-optimal*
409101.bm4 409101bm3 [1, 1, 0, -8354084, 8056799997]  23592960 $$\Gamma_0(N)$$-optimal*
409101.bm2 409101bm4 [1, 1, 0, -34500974, -78013532505] [2, 2] 23592960
409101.bm3 409101bm5 [1, 1, 0, -33522689, -82644538038]  47185920
409101.bm1 409101bm6 [1, 1, 0, -552013739, -4992211246392]  47185920
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 409101bm1.

## Rank

sage: E.rank()

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

## Modular form 409101.2.a.bm

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 