# Properties

 Label 490049c Number of curves $2$ Conductor $490049$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("490049.c1")

sage: E.isogeny_class()

## Elliptic curves in class 490049c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
490049.c2 490049c1 [1, -1, 0, -2626115, -1637360712] [2] 6718464 $$\Gamma_0(N)$$-optimal*
490049.c1 490049c2 [1, -1, 0, -2644000, -1613913477] [2] 13436928 $$\Gamma_0(N)$$-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 490049c1.

## Rank

sage: E.rank()

The elliptic curves in class 490049c have rank $$1$$.

## Modular form 490049.2.a.c

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + 4q^{5} - 3q^{8} - 3q^{9} + 4q^{10} + 4q^{11} - q^{16} - 2q^{17} - 3q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.