Properties

Label 485100.ec
Number of curves $2$
Conductor $485100$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ec1")
 
E.isogeny_class()
 

Elliptic curves in class 485100.ec

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.ec1 485100ec1 \([0, 0, 0, -334425, -75331375]\) \(-3937024/55\) \(-57784924023750000\) \([]\) \(5225472\) \(2.0223\) \(\Gamma_0(N)\)-optimal*
485100.ec2 485100ec2 \([0, 0, 0, 1209075, -377857375]\) \(186050816/166375\) \(-174799395171843750000\) \([]\) \(15676416\) \(2.5716\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 485100.ec1.

Rank

sage: E.rank()
 

The elliptic curves in class 485100.ec have rank \(0\).

Complex multiplication

The elliptic curves in class 485100.ec do not have complex multiplication.

Modular form 485100.2.a.ec

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.