# Properties

 Label 663c Number of curves $2$ Conductor $663$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 663c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
663.b2 663c1 $$[1, 0, 0, -33, -72]$$ $$3981876625/232713$$ $$232713$$ $$$$ $$64$$ $$-0.21668$$ $$\Gamma_0(N)$$-optimal
663.b1 663c2 $$[1, 0, 0, -98, 279]$$ $$104154702625/24649677$$ $$24649677$$ $$$$ $$128$$ $$0.12990$$

## Rank

sage: E.rank()

The elliptic curves in class 663c have rank $$1$$.

## Complex multiplication

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

## Modular form663.2.a.c

sage: E.q_eigenform(10)

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