# Properties

 Label 1089g Number of curves 4 Conductor 1089 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1089.j1")

sage: E.isogeny_class()

## Elliptic curves in class 1089g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1089.j3 1089g1 [1, -1, 0, -7101, -226616]  1440 $$\Gamma_0(N)$$-optimal
1089.j2 1089g2 [1, -1, 0, -12546, 173047] [2, 2] 2880
1089.j1 1089g3 [1, -1, 0, -159561, 24548134]  5760
1089.j4 1089g4 [1, -1, 0, 47349, 1311052]  5760

## Rank

sage: E.rank()

The elliptic curves in class 1089g have rank $$1$$.

## Modular form1089.2.a.j

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + 2q^{5} - 4q^{7} - 3q^{8} + 2q^{10} + 2q^{13} - 4q^{14} - q^{16} - 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 