# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 8550.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8550.x1 8550x3 $$[1, -1, 1, -139730, 20109647]$$ $$26487576322129/44531250$$ $$507238769531250$$ $$$$ $$49152$$ $$1.7169$$
8550.x2 8550x2 $$[1, -1, 1, -11480, 102647]$$ $$14688124849/8122500$$ $$92520351562500$$ $$[2, 2]$$ $$24576$$ $$1.3704$$
8550.x3 8550x1 $$[1, -1, 1, -6980, -221353]$$ $$3301293169/22800$$ $$259706250000$$ $$$$ $$12288$$ $$1.0238$$ $$\Gamma_0(N)$$-optimal
8550.x4 8550x4 $$[1, -1, 1, 44770, 777647]$$ $$871257511151/527800050$$ $$-6011972444531250$$ $$$$ $$49152$$ $$1.7169$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 8550.x do not have complex multiplication.

## 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 LMFDB 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 LMFDB labels. 