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Properties

Label 481650ge
Number of curves $4$
Conductor $481650$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ge1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 481650ge have rank \(0\).

Complex multiplication

The elliptic curves in class 481650ge do not have complex multiplication.

Modular form 481650.2.a.ge

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + 2 q^{7} + q^{8} + q^{9} - 6 q^{11} - q^{12} + 2 q^{14} + q^{16} + 6 q^{17} + q^{18} - q^{19} + O(q^{20})\) 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 481650ge

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481650.ge3 481650ge1 \([1, 1, 1, -109305063, 443846474781]\) \(-1914980734749238129/20440940544000\) \(-1541633059160064000000000\) \([2]\) \(149299200\) \(3.4577\) \(\Gamma_0(N)\)-optimal*
481650.ge2 481650ge2 \([1, 1, 1, -1753337063, 28257579850781]\) \(7903870428425797297009/886464000000\) \(66856131459000000000000\) \([2]\) \(298598400\) \(3.8042\) \(\Gamma_0(N)\)-optimal*
481650.ge4 481650ge3 \([1, 1, 1, 361190937, 2310807050781]\) \(69096190760262356111/70568821500000000\) \(-5322222230243648437500000000\) \([2]\) \(447897600\) \(4.0070\)  
481650.ge1 481650ge4 \([1, 1, 1, -1957151063, 21279481294781]\) \(10993009831928446009969/3767761230468750000\) \(284160372141838073730468750000\) \([2]\) \(895795200\) \(4.3536\)  
*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 481650ge1.