# Properties

 Label 8550x Number of curves $4$ Conductor $8550$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("8550.x1")

sage: E.isogeny_class()

## Elliptic curves in class 8550x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8550.x3 8550x1 [1, -1, 1, -6980, -221353]  12288 $$\Gamma_0(N)$$-optimal
8550.x2 8550x2 [1, -1, 1, -11480, 102647] [2, 2] 24576
8550.x1 8550x3 [1, -1, 1, -139730, 20109647]  49152
8550.x4 8550x4 [1, -1, 1, 44770, 777647]  49152

## Rank

sage: E.rank()

The elliptic curves in class 8550x have rank $$1$$.

## Modular form8550.2.a.x

sage: E.q_eigenform(10)

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

## 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. 