Properties

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

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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 \([1, 0, 1, -163980, -22341014]\) \(4144806984356137/568114785504\) \(66838136399760096\) \([2]\) \(92160\) \(1.9553\)  
6762.r2 6762n1 \([1, 0, 1, 16340, -1856662]\) \(4101378352343/15049939968\) \(-1770610387295232\) \([2]\) \(46080\) \(1.6087\) \(\Gamma_0(N)\)-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 form 6762.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{8} + q^{9} - 2 q^{10} + q^{12} - 4 q^{13} + 2 q^{15} + q^{16} - q^{18} - 6 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.