# Properties

 Label 64.a Number of curves $4$ Conductor $64$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 64.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
64.a1 64a2 $$[0, 0, 0, -44, -112]$$ $$287496$$ $$32768$$ $$$$ $$4$$ $$-0.27081$$   $$-16$$
64.a2 64a3 $$[0, 0, 0, -44, 112]$$ $$287496$$ $$32768$$ $$$$ $$4$$ $$-0.27081$$   $$-16$$
64.a3 64a1 $$[0, 0, 0, -4, 0]$$ $$1728$$ $$4096$$ $$[2, 2]$$ $$2$$ $$-0.61739$$ $$\Gamma_0(N)$$-optimal $$-4$$
64.a4 64a4 $$[0, 0, 0, 1, 0]$$ $$1728$$ $$-64$$ $$$$ $$4$$ $$-0.96396$$   $$-4$$

## Rank

sage: E.rank()

The elliptic curves in class 64.a have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 64.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form64.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{5} - 3q^{9} - 6q^{13} + 2q^{17} + O(q^{20})$$ ## 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. 