# Properties

 Label 1989e Number of curves $6$ Conductor $1989$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1989.e1")

sage: E.isogeny_class()

## Elliptic curves in class 1989e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1989.e4 1989e1 [1, -1, 0, -4851, -128840] [2] 1024 $$\Gamma_0(N)$$-optimal
1989.e3 1989e2 [1, -1, 0, -4896, -126293] [2, 2] 2048
1989.e2 1989e3 [1, -1, 0, -12501, 368032] [2, 2] 4096
1989.e5 1989e4 [1, -1, 0, 1989, -458150] [2] 4096
1989.e1 1989e5 [1, -1, 0, -181566, 29819155] [2] 8192
1989.e6 1989e6 [1, -1, 0, 34884, 2462449] [2] 8192

## Rank

sage: E.rank()

The elliptic curves in class 1989e have rank $$1$$.

## Modular form1989.2.a.e

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.