# Properties

 Label 4650.x Number of curves $2$ Conductor $4650$ CM no Rank $1$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 4650.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4650.x1 4650bd2 $$[1, 1, 1, -5461188, 4909955781]$$ $$1152829477932246539641/3188367360$$ $$49818240000000$$ $$$$ $$99840$$ $$2.2864$$
4650.x2 4650bd1 $$[1, 1, 1, -341188, 76675781]$$ $$-281115640967896441/468084326400$$ $$-7313817600000000$$ $$$$ $$49920$$ $$1.9399$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4650.x have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4650.x do not have complex multiplication.

## Modular form4650.2.a.x

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} - 4 q^{7} + q^{8} + q^{9} + 2 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + q^{16} + q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 