# Properties

 Label 490.f Number of curves $2$ Conductor $490$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("490.f1")

sage: E.isogeny_class()

## Elliptic curves in class 490.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
490.f1 490g2 [1, 0, 0, -1191, 15721]  320
490.f2 490g1 [1, 0, 0, -71, 265]  160 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 490.f have rank $$1$$.

## Modular form490.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} - 2q^{12} - 2q^{13} + 2q^{15} + q^{16} - 8q^{17} + q^{18} + 6q^{19} + O(q^{20})$$

## 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. 