# Properties

 Label 494190bn Number of curves $4$ Conductor $494190$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("494190.bn1")

sage: E.isogeny_class()

## Elliptic curves in class 494190bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
494190.bn4 494190bn1 [1, -1, 0, -26064, -2757200] [2] 3932160 $$\Gamma_0(N)$$-optimal*
494190.bn3 494190bn2 [1, -1, 0, -494244, -133566692] [2, 2] 7864320 $$\Gamma_0(N)$$-optimal*
494190.bn2 494190bn3 [1, -1, 0, -572274, -88512170] [2] 15728640 $$\Gamma_0(N)$$-optimal*
494190.bn1 494190bn4 [1, -1, 0, -7907094, -8556046862] [2] 15728640
*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 494190bn1.

## Rank

sage: E.rank()

The elliptic curves in class 494190bn have rank $$1$$.

## Modular form 494190.2.a.bn

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} - 4q^{11} - 2q^{13} + 4q^{14} + q^{16} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.