Properties

Label 510f
Number of curves $4$
Conductor $510$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("510.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 510f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
510.f4 510f1 [1, 0, 0, 4, 0] [2] 32 \(\Gamma_0(N)\)-optimal
510.f3 510f2 [1, 0, 0, -16, -4] [2, 2] 64  
510.f1 510f3 [1, 0, 0, -186, -990] [2] 128  
510.f2 510f4 [1, 0, 0, -166, 806] [2] 128  

Rank

sage: E.rank()
 

The elliptic curves in class 510f have rank \(0\).

Modular form 510.2.a.f

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.