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Properties

Label 488400.fr
Number of curves $2$
Conductor $488400$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 488400.fr have rank \(0\).

Complex multiplication

The elliptic curves in class 488400.fr do not have complex multiplication.

Modular form 488400.2.a.fr

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{7} + q^{9} + q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 488400.fr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
488400.fr1 488400fr2 \([0, 1, 0, -406208, 99489588]\) \(115821777093625/31804608\) \(2035494912000000\) \([2]\) \(3981312\) \(1.9199\) \(\Gamma_0(N)\)-optimal*
488400.fr2 488400fr1 \([0, 1, 0, -22208, 1953588]\) \(-18927429625/15003648\) \(-960233472000000\) \([2]\) \(1990656\) \(1.5733\) \(\Gamma_0(N)\)-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 488400.fr1.