Properties

Label 466578.ce
Number of curves $6$
Conductor $466578$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("466578.ce1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 466578.ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
466578.ce1 466578ce5 [1, -1, 0, -18461096586, 965455066178280] [2] 830472192 \(\Gamma_0(N)\)-optimal*
466578.ce2 466578ce4 [1, -1, 0, -4132486206, -102246673939740] [2] 415236096  
466578.ce3 466578ce3 [1, -1, 0, -1183713246, 14262548301252] [2, 2] 415236096 \(\Gamma_0(N)\)-optimal*
466578.ce4 466578ce2 [1, -1, 0, -269220366, -1454840827308] [2, 2] 207618048 \(\Gamma_0(N)\)-optimal*
466578.ce5 466578ce1 [1, -1, 0, 29389554, -125608629420] [2] 103809024 \(\Gamma_0(N)\)-optimal*
466578.ce6 466578ce6 [1, -1, 0, 1461784014, 68971960737504] [2] 830472192  
*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 466578.ce5.

Rank

sage: E.rank()
 

The elliptic curves in class 466578.ce have rank \(0\).

Modular form 466578.2.a.ce

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.