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Properties

Label 462550d
Number of curves $4$
Conductor $462550$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462550d have rank \(1\).

Complex multiplication

The elliptic curves in class 462550d do not have complex multiplication.

Modular form 462550.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - 2 q^{7} - q^{8} + q^{9} - q^{11} - 2 q^{12} + 4 q^{13} + 2 q^{14} + q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 462550d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462550.d4 462550d1 \([1, 0, 1, 2890499, 57603378648]\) \(287365339799/154396000000\) \(-1434974085454937500000000\) \([2]\) \(69672960\) \(3.3143\) \(\Gamma_0(N)\)-optimal*
462550.d2 462550d2 \([1, 0, 1, -207359501, 1121047878648]\) \(106093191228100201/2979765602000\) \(27694282362237566281250000\) \([2]\) \(139345920\) \(3.6609\) \(\Gamma_0(N)\)-optimal*
462550.d3 462550d3 \([1, 0, 1, -26018876, -1556753940102]\) \(-209595169258201/112524368281600\) \(-1045814350542005862400000000\) \([2]\) \(209018880\) \(3.8636\)  
462550.d1 462550d4 \([1, 0, 1, -2178978876, -38742679060102]\) \(123104735252886403801/1474019775303680\) \(13699708405715761267520000000\) \([2]\) \(418037760\) \(4.2102\)  
*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 462550d1.