# Properties

 Label 89280.ed Number of curves $6$ Conductor $89280$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("89280.ed1")

sage: E.isogeny_class()

## Elliptic curves in class 89280.ed

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
89280.ed1 89280ca6 [0, 0, 0, -177131532, 907385702384] [2] 6291456
89280.ed2 89280ca4 [0, 0, 0, -11070732, 14177871344] [2, 2] 3145728
89280.ed3 89280ca5 [0, 0, 0, -10897932, 14641873904] [2] 6291456
89280.ed4 89280ca3 [0, 0, 0, -2131212, -933677584] [2] 3145728
89280.ed5 89280ca2 [0, 0, 0, -702732, 214248944] [2, 2] 1572864
89280.ed6 89280ca1 [0, 0, 0, 34548, 14003696] [2] 786432 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 89280.ed have rank $$1$$.

## Modular form 89280.2.a.ed

sage: E.q_eigenform(10)

$$q + q^{5} - 4q^{11} - 6q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.