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Properties

Label 400710.x
Number of curves $6$
Conductor $400710$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("400710.x1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 400710.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
400710.x1 400710x4 [1, 0, 1, -123102813, -525724902344] [2] 47775744  
400710.x2 400710x5 [1, 0, 1, -109428133, 438672429128] [2] 95551488 \(\Gamma_0(N)\)-optimal*
400710.x3 400710x3 [1, 0, 1, -10586333, -1489874632] [2, 2] 47775744 \(\Gamma_0(N)\)-optimal*
400710.x4 400710x2 [1, 0, 1, -7698333, -8205052232] [2, 2] 23887872 \(\Gamma_0(N)\)-optimal*
400710.x5 400710x1 [1, 0, 1, -305053, -223267144] [2] 11943936 \(\Gamma_0(N)\)-optimal*
400710.x6 400710x6 [1, 0, 1, 42047467, -11869259992] [2] 95551488  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 400710.x5.

Rank

sage: E.rank()
 

The elliptic curves in class 400710.x have rank \(0\).

Modular form 400710.2.a.x

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.