# Properties

 Label 663.c Number of curves $2$ Conductor $663$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 663.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
663.c1 663a2 $$[1, 1, 0, -327, -900]$$ $$3885442650361/1996623837$$ $$1996623837$$ $$$$ $$576$$ $$0.47674$$
663.c2 663a1 $$[1, 1, 0, -262, -1745]$$ $$2000852317801/2094417$$ $$2094417$$ $$$$ $$288$$ $$0.13017$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 663.c have rank $$0$$.

## Complex multiplication

The elliptic curves in class 663.c do not have complex multiplication.

## Modular form663.2.a.c

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - 4q^{5} - q^{6} + 2q^{7} - 3q^{8} + q^{9} - 4q^{10} + 6q^{11} + q^{12} - q^{13} + 2q^{14} + 4q^{15} - q^{16} + q^{17} + q^{18} + 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 