# Properties

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

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 663b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
663.a4 663b1 $$[1, 1, 1, -539, 4592]$$ $$17319700013617/25857$$ $$25857$$ $$[4]$$ $$128$$ $$0.11544$$ $$\Gamma_0(N)$$-optimal
663.a3 663b2 $$[1, 1, 1, -544, 4496]$$ $$17806161424897/668584449$$ $$668584449$$ $$[2, 4]$$ $$256$$ $$0.46201$$
663.a2 663b3 $$[1, 1, 1, -1389, -14094]$$ $$296380748763217/92608836489$$ $$92608836489$$ $$[2, 2]$$ $$512$$ $$0.80858$$
663.a5 663b4 $$[1, 1, 1, 221, 17042]$$ $$1193377118543/124806800313$$ $$-124806800313$$ $$[4]$$ $$512$$ $$0.80858$$
663.a1 663b5 $$[1, 1, 1, -20174, -1111138]$$ $$908031902324522977/161726530797$$ $$161726530797$$ $$[2]$$ $$1024$$ $$1.1552$$
663.a6 663b6 $$[1, 1, 1, 3876, -89910]$$ $$6439735268725823/7345472585373$$ $$-7345472585373$$ $$[2]$$ $$1024$$ $$1.1552$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form663.2.a.b

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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.