Properties

Label 471510ep
Number of curves $6$
Conductor $471510$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("471510.ep1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 471510ep

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
471510.ep6 471510ep1 [1, -1, 1, 91228, 60067271] [2] 7864320 \(\Gamma_0(N)\)-optimal*
471510.ep5 471510ep2 [1, -1, 1, -1855652, 919809479] [2, 2] 15728640 \(\Gamma_0(N)\)-optimal*
471510.ep2 471510ep3 [1, -1, 1, -29233652, 60844775879] [2, 2] 31457280 \(\Gamma_0(N)\)-optimal*
471510.ep4 471510ep4 [1, -1, 1, -5627732, -4005018169] [2] 31457280  
471510.ep1 471510ep5 [1, -1, 1, -467737952, 3893723161319] [2] 62914560 \(\Gamma_0(N)\)-optimal*
471510.ep3 471510ep6 [1, -1, 1, -28777352, 62835704039] [2] 62914560  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 471510ep1.

Rank

sage: E.rank()
 

The elliptic curves in class 471510ep have rank \(1\).

Modular form 471510.2.a.ep

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 4q^{11} + q^{16} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.