# Properties

 Label 40.a Number of curves $4$ Conductor $40$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 40.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40.a1 40a2 $$[0, 0, 0, -107, -426]$$ $$132304644/5$$ $$5120$$ $$$$ $$4$$ $$-0.20221$$
40.a2 40a1 $$[0, 0, 0, -7, -6]$$ $$148176/25$$ $$6400$$ $$[2, 2]$$ $$2$$ $$-0.54879$$ $$\Gamma_0(N)$$-optimal
40.a3 40a3 $$[0, 0, 0, -2, 1]$$ $$55296/5$$ $$80$$ $$$$ $$4$$ $$-0.89536$$
40.a4 40a4 $$[0, 0, 0, 13, -34]$$ $$237276/625$$ $$-640000$$ $$$$ $$4$$ $$-0.20221$$

## Rank

sage: E.rank()

The elliptic curves in class 40.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 40.a do not have complex multiplication.

## Modular form40.2.a.a

sage: E.q_eigenform(10)

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