# Properties

 Label 288990ff Number of curves $4$ Conductor $288990$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("288990.ff1")

sage: E.isogeny_class()

## Elliptic curves in class 288990ff

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
288990.ff4 288990ff1 [1, -1, 1, -15242, -1232071] [2] 1769472 $$\Gamma_0(N)$$-optimal
288990.ff3 288990ff2 [1, -1, 1, -289022, -59711479] [2, 2] 3538944
288990.ff2 288990ff3 [1, -1, 1, -334652, -39561271] [2] 7077888
288990.ff1 288990ff4 [1, -1, 1, -4623872, -3825829159] [2] 7077888

## Rank

sage: E.rank()

The elliptic curves in class 288990ff have rank $$1$$.

## Modular form 288990.2.a.ff

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} - 4q^{7} + q^{8} + q^{10} - 4q^{11} - 4q^{14} + q^{16} + 2q^{17} + q^{19} + O(q^{20})$$

## 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.