# Properties

 Label 288.d Number of curves $4$ Conductor $288$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 288.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
288.d1 288d2 $$[0, 0, 0, -99, -378]$$ $$287496$$ $$373248$$ $$[2]$$ $$32$$ $$-0.068080$$   $$-16$$
288.d2 288d3 $$[0, 0, 0, -99, 378]$$ $$287496$$ $$373248$$ $$[2]$$ $$32$$ $$-0.068080$$   $$-16$$
288.d3 288d1 $$[0, 0, 0, -9, 0]$$ $$1728$$ $$46656$$ $$[2, 2]$$ $$16$$ $$-0.41465$$ $$\Gamma_0(N)$$-optimal $$-4$$
288.d4 288d4 $$[0, 0, 0, 36, 0]$$ $$1728$$ $$-2985984$$ $$[2]$$ $$32$$ $$-0.068080$$   $$-4$$

## Rank

sage: E.rank()

The elliptic curves in class 288.d have rank $$0$$.

## Complex multiplication

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

## Modular form288.2.a.d

sage: E.q_eigenform(10)

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

## 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.