Properties

Label 770f
Number of curves 4
Conductor 770
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("770.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 770f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
770.f3 770f1 [1, 0, 0, -56, 3136] [6] 576 \(\Gamma_0(N)\)-optimal
770.f2 770f2 [1, 0, 0, -3576, 81280] [6] 1152  
770.f4 770f3 [1, 0, 0, 504, -84560] [2] 1728  
770.f1 770f4 [1, 0, 0, -26116, -1580604] [2] 3456  

Rank

sage: E.rank()
 

The elliptic curves in class 770f have rank \(1\).

Modular form 770.2.a.f

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.