# Properties

 Label 47040ej Number of curves $2$ Conductor $47040$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47040.be1")

sage: E.isogeny_class()

## Elliptic curves in class 47040ej

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.be1 47040ej1 [0, -1, 0, -961, 4705] [2] 36864 $$\Gamma_0(N)$$-optimal
47040.be2 47040ej2 [0, -1, 0, 3519, 32481] [2] 73728

## Rank

sage: E.rank()

The elliptic curves in class 47040ej have rank $$1$$.

## Modular form 47040.2.a.be

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} + 2q^{11} - 2q^{13} + q^{15} - 4q^{17} + O(q^{20})$$

## 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.