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Properties

Label 722.b
Number of curves $2$
Conductor $722$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 722.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
722.b1 722c2 \([1, 0, 1, -25278, 1710222]\) \(-37966934881/4952198\) \(-232980517796438\) \([]\) \(3600\) \(1.4900\)  
722.b2 722c1 \([1, 0, 1, -8, -8138]\) \(-1/608\) \(-28603895648\) \([]\) \(720\) \(0.68529\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 722.b have rank \(0\).

Complex multiplication

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

Modular form 722.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 4 q^{5} - q^{6} + 3 q^{7} - q^{8} - 2 q^{9} + 4 q^{10} + 2 q^{11} + q^{12} + q^{13} - 3 q^{14} - 4 q^{15} + q^{16} + 3 q^{17} + 2 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.