# Properties

 Label 139650.ij Number of curves $2$ Conductor $139650$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ij1")

sage: E.isogeny_class()

## Elliptic curves in class 139650.ij

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.ij1 139650by1 $$[1, 0, 0, -547478, -213208668]$$ $$-2569823930905/1292464512$$ $$-9127230871271587200$$ $$[]$$ $$3048192$$ $$2.3431$$ $$\Gamma_0(N)$$-optimal
139650.ij2 139650by2 $$[1, 0, 0, 4314547, 2673861777]$$ $$1257792236741495/1165133611008$$ $$-8228035172193848524800$$ $$[]$$ $$9144576$$ $$2.8924$$

## Rank

sage: E.rank()

The elliptic curves in class 139650.ij have rank $$0$$.

## Complex multiplication

The elliptic curves in class 139650.ij do not have complex multiplication.

## Modular form 139650.2.a.ij

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + q^{12} + 5q^{13} + q^{16} + 6q^{17} + q^{18} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.