# Properties

 Label 576.e Number of curves $4$ Conductor $576$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 576.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
576.e1 576a4 $$[0, 0, 0, -540, -4752]$$ $$54000$$ $$322486272$$ $$[2]$$ $$192$$ $$0.42704$$   $$-12$$
576.e2 576a2 $$[0, 0, 0, -60, 176]$$ $$54000$$ $$442368$$ $$[2]$$ $$64$$ $$-0.12227$$   $$-12$$
576.e3 576a3 $$[0, 0, 0, 0, -216]$$ $$0$$ $$-20155392$$ $$[2]$$ $$96$$ $$0.080464$$   $$-3$$
576.e4 576a1 $$[0, 0, 0, 0, 8]$$ $$0$$ $$-27648$$ $$[2]$$ $$32$$ $$-0.46884$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 576.e have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 576.e has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form576.2.a.e

sage: E.q_eigenform(10)

$$q - 4 q^{7} - 2 q^{13} - 8 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.