# Properties

 Label 288.e Number of curves $2$ Conductor $288$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 288.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
288.e1 288e2 $$[0, 0, 0, -108, 0]$$ $$1728$$ $$80621568$$ $$$$ $$96$$ $$0.20657$$   $$-4$$
288.e2 288e1 $$[0, 0, 0, 27, 0]$$ $$1728$$ $$-1259712$$ $$$$ $$48$$ $$-0.14000$$ $$\Gamma_0(N)$$-optimal $$-4$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form288.2.a.e

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. 