# Properties

 Label 32a Number of curves $4$ Conductor $32$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 32a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
32.a4 32a1 $$[0, 0, 0, 4, 0]$$ $$1728$$ $$-4096$$ $$$$ $$1$$ $$-0.61739$$ $$\Gamma_0(N)$$-optimal $$-4$$
32.a3 32a2 $$[0, 0, 0, -1, 0]$$ $$1728$$ $$64$$ $$[2, 2]$$ $$2$$ $$-0.96396$$   $$-4$$
32.a1 32a3 $$[0, 0, 0, -11, -14]$$ $$287496$$ $$512$$ $$$$ $$4$$ $$-0.61739$$   $$-16$$
32.a2 32a4 $$[0, 0, 0, -11, 14]$$ $$287496$$ $$512$$ $$$$ $$4$$ $$-0.61739$$   $$-16$$

## Rank

sage: E.rank()

The elliptic curves in class 32a have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 32a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form32.2.a.a

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels. 