# Properties

 Label 663.a Number of curves $6$ Conductor $663$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("663.a1")

sage: E.isogeny_class()

## Elliptic curves in class 663.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
663.a1 663b5 [1, 1, 1, -20174, -1111138] [2] 1024
663.a2 663b3 [1, 1, 1, -1389, -14094] [2, 2] 512
663.a3 663b2 [1, 1, 1, -544, 4496] [2, 4] 256
663.a4 663b1 [1, 1, 1, -539, 4592] [4] 128 $$\Gamma_0(N)$$-optimal
663.a5 663b4 [1, 1, 1, 221, 17042] [4] 512
663.a6 663b6 [1, 1, 1, 3876, -89910] [2] 1024

## Rank

sage: E.rank()

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

## Modular form663.2.a.a

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.