# Properties

 Label 416955ce Number of curves $6$ Conductor $416955$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("416955.ce1")

sage: E.isogeny_class()

## Elliptic curves in class 416955ce

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
416955.ce4 416955ce1 [1, 0, 1, -95673, -11398049] [2] 1769472 $$\Gamma_0(N)$$-optimal*
416955.ce3 416955ce2 [1, 0, 1, -97478, -10946077] [2, 2] 3538944 $$\Gamma_0(N)$$-optimal*
416955.ce2 416955ce3 [1, 0, 1, -315883, 55361681] [2, 2] 7077888 $$\Gamma_0(N)$$-optimal*
416955.ce5 416955ce4 [1, 0, 1, 92047, -48320407] [2] 7077888
416955.ce1 416955ce5 [1, 0, 1, -4783258, 4025964581] [2] 14155776 $$\Gamma_0(N)$$-optimal*
416955.ce6 416955ce6 [1, 0, 1, 657012, 329328913] [2] 14155776
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 416955ce1.

## Rank

sage: E.rank()

The elliptic curves in class 416955ce have rank $$1$$.

## Modular form 416955.2.a.ce

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - q^{7} - 3q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + 2q^{13} - q^{14} + q^{15} - q^{16} - 6q^{17} + q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.