# Properties

 Label 576.c Number of curves $4$ Conductor $576$ CM $$\Q(\sqrt{-1})$$ Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 576.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
576.c1 576h3 $$[0, 0, 0, -396, -3024]$$ $$287496$$ $$23887872$$ $$$$ $$128$$ $$0.27849$$   $$-16$$
576.c2 576h4 $$[0, 0, 0, -396, 3024]$$ $$287496$$ $$23887872$$ $$$$ $$128$$ $$0.27849$$   $$-16$$
576.c3 576h2 $$[0, 0, 0, -36, 0]$$ $$1728$$ $$2985984$$ $$[2, 2]$$ $$64$$ $$-0.068080$$   $$-4$$
576.c4 576h1 $$[0, 0, 0, 9, 0]$$ $$1728$$ $$-46656$$ $$$$ $$32$$ $$-0.41465$$ $$\Gamma_0(N)$$-optimal $$-4$$

## Rank

sage: E.rank()

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

## Complex multiplication

Each elliptic curve in class 576.c has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form576.2.a.c

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. 