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Properties

Label 458640.fp
Number of curves $4$
Conductor $458640$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640.fp have rank \(0\).

Complex multiplication

The elliptic curves in class 458640.fp do not have complex multiplication.

Modular form 458640.2.a.fp

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 458640.fp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.fp1 458640fp4 \([0, 0, 0, -1112975080203, -451935309840037702]\) \(434014578033107719741685694649/103121648659575000\) \(36226432191105422371123200000\) \([2]\) \(4246732800\) \(5.2986\)  
458640.fp2 458640fp2 \([0, 0, 0, -69569080203, -7059754388437702]\) \(105997782562506306791694649/51649016225625000000\) \(18144197734964908239360000000000\) \([2, 2]\) \(2123366400\) \(4.9520\)  
458640.fp3 458640fp3 \([0, 0, 0, -57978330123, -9489317012356678]\) \(-61354313914516350666047929/75227254486083984375000\) \(-26427186424837209375000000000000000\) \([2]\) \(4246732800\) \(5.2986\)  
458640.fp4 458640fp1 \([0, 0, 0, -5080627083, -70611918507718]\) \(41285728533151645510969/17760741842188800000\) \(6239313649200855145041100800000\) \([2]\) \(1061683200\) \(4.6054\) \(\Gamma_0(N)\)-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 458640.fp1.