Properties

Label 466578cj
Number of curves $4$
Conductor $466578$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cj1")
 
E.isogeny_class()
 

Elliptic curves in class 466578cj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466578.cj4 466578cj1 \([1, -1, 0, -139744971, 1037250589045]\) \(-23771111713777/22848457968\) \(-290094623322307610514471792\) \([2]\) \(194641920\) \(3.7738\) \(\Gamma_0(N)\)-optimal*
466578.cj3 466578cj2 \([1, -1, 0, -2607942591, 51246807493657]\) \(154502321244119857/55101928644\) \(699599651613296661840902436\) \([2, 2]\) \(389283840\) \(4.1204\) \(\Gamma_0(N)\)-optimal*
466578.cj1 466578cj3 \([1, -1, 0, -41723509221, 3280354179500047]\) \(632678989847546725777/80515134\) \(1022257497735180307455246\) \([2]\) \(778567680\) \(4.4670\) \(\Gamma_0(N)\)-optimal*
466578.cj2 466578cj4 \([1, -1, 0, -2983537881, 35521459010995]\) \(231331938231569617/90942310746882\) \(1154645771593235997718605687858\) \([2]\) \(778567680\) \(4.4670\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 466578cj1.

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 466578cj do not have complex multiplication.

Modular form 466578.2.a.cj

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

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.