# Properties

 Label 66139.a Number of curves $3$ Conductor $66139$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 66139.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
66139.a1 66139a3 $$[0, 1, 1, -2678049, 1685955405]$$ $$-50357871050752/19$$ $$-801430139179$$ $$[]$$ $$620136$$ $$2.0722$$
66139.a2 66139a2 $$[0, 1, 1, -32489, 2387160]$$ $$-89915392/6859$$ $$-289316280243619$$ $$[]$$ $$206712$$ $$1.5229$$
66139.a3 66139a1 $$[0, 1, 1, 2321, 2675]$$ $$32768/19$$ $$-801430139179$$ $$[]$$ $$68904$$ $$0.97360$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 66139.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 66139.a do not have complex multiplication.

## Modular form 66139.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{4} + 3q^{5} - q^{7} + q^{9} - 3q^{11} + 4q^{12} + 4q^{13} - 6q^{15} + 4q^{16} - 3q^{17} + q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 