Properties

Label 735a
Number of curves 4
Conductor 735
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("735.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 735a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
735.f3 735a1 [1, 1, 0, -123, -552] [2] 192 \(\Gamma_0(N)\)-optimal
735.f2 735a2 [1, 1, 0, -368, 1947] [2, 2] 384  
735.f1 735a3 [1, 1, 0, -5513, 155268] [2] 768  
735.f4 735a4 [1, 1, 0, 857, 13462] [2] 768  

Rank

sage: E.rank()
 

The elliptic curves in class 735a have rank \(0\).

Modular form 735.2.a.f

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.