# Properties

 Label 6762.r Number of curves $2$ Conductor $6762$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 6762.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6762.r1 6762n2 $$[1, 0, 1, -163980, -22341014]$$ $$4144806984356137/568114785504$$ $$66838136399760096$$ $$$$ $$92160$$ $$1.9553$$
6762.r2 6762n1 $$[1, 0, 1, 16340, -1856662]$$ $$4101378352343/15049939968$$ $$-1770610387295232$$ $$$$ $$46080$$ $$1.6087$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 6762.r have rank $$1$$.

## Complex multiplication

The elliptic curves in class 6762.r do not have complex multiplication.

## Modular form6762.2.a.r

sage: E.q_eigenform(10)

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

## 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. 