# Properties

 Label 39.a Number of curves $4$ Conductor $39$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 39.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
39.a1 39a2 $$[1, 1, 0, -69, -252]$$ $$37159393753/1053$$ $$1053$$ $$$$ $$4$$ $$-0.31837$$
39.a2 39a3 $$[1, 1, 0, -19, 22]$$ $$822656953/85683$$ $$85683$$ $$$$ $$4$$ $$-0.31837$$
39.a3 39a1 $$[1, 1, 0, -4, -5]$$ $$10218313/1521$$ $$1521$$ $$[2, 2]$$ $$2$$ $$-0.66494$$ $$\Gamma_0(N)$$-optimal
39.a4 39a4 $$[1, 1, 0, 1, 0]$$ $$12167/39$$ $$-39$$ $$$$ $$4$$ $$-1.0115$$

## Rank

sage: E.rank()

The elliptic curves in class 39.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 39.a do not have complex multiplication.

## Modular form39.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - 4q^{7} - 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + q^{13} - 4q^{14} - 2q^{15} - q^{16} + 2q^{17} + q^{18} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 