Properties

Label 418275.cg
Number of curves $6$
Conductor $418275$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("418275.cg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 418275.cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
418275.cg1 418275cg4 [1, -1, 0, -261004392, -1622939671859] [2] 44040192  
418275.cg2 418275cg5 [1, -1, 0, -114418017, 456186144766] [2] 88080384 \(\Gamma_0(N)\)-optimal*
418275.cg3 418275cg3 [1, -1, 0, -18024642, -19707947609] [2, 2] 44040192 \(\Gamma_0(N)\)-optimal*
418275.cg4 418275cg2 [1, -1, 0, -16313517, -25352948984] [2, 2] 22020096 \(\Gamma_0(N)\)-optimal*
418275.cg5 418275cg1 [1, -1, 0, -913392, -481747109] [2] 11010048 \(\Gamma_0(N)\)-optimal*
418275.cg6 418275cg6 [1, -1, 0, 50990733, -134342485484] [2] 88080384  
*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 418275.cg5.

Rank

sage: E.rank()
 

The elliptic curves in class 418275.cg have rank \(0\).

Modular form 418275.2.a.cg

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

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 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.