# Properties

 Label 288.a Number of curves $2$ Conductor $288$ CM $$\Q(\sqrt{-1})$$ Rank $1$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 288.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
288.a1 288a2 $$[0, 0, 0, -12, 0]$$ $$1728$$ $$110592$$ $$$$ $$32$$ $$-0.34273$$   $$-4$$
288.a2 288a1 $$[0, 0, 0, 3, 0]$$ $$1728$$ $$-1728$$ $$$$ $$16$$ $$-0.68931$$ $$\Gamma_0(N)$$-optimal $$-4$$

## Rank

sage: E.rank()

The elliptic curves in class 288.a have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 288.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form288.2.a.a

sage: E.q_eigenform(10)

$$q - 4q^{5} - 6q^{13} - 8q^{17} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 