Properties

Label 714i
Number of curves $3$
Conductor $714$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("714.i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 714i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
714.i3 714i1 [1, 0, 0, 108, 11664] [9] 1080 \(\Gamma_0(N)\)-optimal
714.i2 714i2 [1, 0, 0, -972, -315144] [3] 3240  
714.i1 714i3 [1, 0, 0, -381702, -90803346] [] 9720  

Rank

sage: E.rank()
 

The elliptic curves in class 714i have rank \(0\).

Modular form 714.2.a.i

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - 3q^{10} + 3q^{11} + q^{12} + 5q^{13} + q^{14} - 3q^{15} + q^{16} - q^{17} + q^{18} + 2q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.