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Properties

Label 400554.fr
Number of curves $2$
Conductor $400554$
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 400554.fr have rank \(0\).

Complex multiplication

The elliptic curves in class 400554.fr do not have complex multiplication.

Modular form 400554.2.a.fr

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 4 q^{5} - q^{7} + q^{8} + 4 q^{10} - q^{11} - q^{14} + q^{16} + 4 q^{19} + O(q^{20})\) 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 400554.fr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400554.fr1 400554fr2 \([1, -1, 1, -10538873, -13078725991]\) \(198631853028508563/1517849044096\) \(989204022903183550848\) \([2]\) \(37158912\) \(2.8580\)  
400554.fr2 400554fr1 \([1, -1, 1, -1105913, 108552089]\) \(229524442504083/125055451136\) \(81500433676775866368\) \([2]\) \(18579456\) \(2.5114\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 400554.fr1.