Properties

Label 722.c
Number of curves $2$
Conductor $722$
CM no
Rank $1$
Graph

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

Elliptic curves in class 722.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
722.c1 722a2 \([1, 0, 1, -33581, -2375576]\) \(-246579625/512\) \(-8695584276992\) \([]\) \(2052\) \(1.3686\)  
722.c2 722a1 \([1, 0, 1, 714, -16080]\) \(2375/8\) \(-135868504328\) \([3]\) \(684\) \(0.81932\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 722.c have rank \(1\).

Complex multiplication

The elliptic curves in class 722.c do not have complex multiplication.

Modular form 722.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - 4q^{7} - q^{8} - 2q^{9} + 3q^{11} + q^{12} + 2q^{13} + 4q^{14} + q^{16} - 6q^{17} + 2q^{18} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.