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Properties

Label 436800ge
Number of curves $4$
Conductor $436800$
CM no
Rank $2$
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 436800ge have rank \(2\).

Complex multiplication

The elliptic curves in class 436800ge do not have complex multiplication.

Modular form 436800.2.a.ge

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 436800ge

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
436800.ge3 436800ge1 \([0, -1, 0, -3108, -65538]\) \(3321287488/7371\) \(7371000000\) \([2]\) \(393216\) \(0.77607\) \(\Gamma_0(N)\)-optimal*
436800.ge2 436800ge2 \([0, -1, 0, -4233, -12663]\) \(131096512/74529\) \(4769856000000\) \([2, 2]\) \(786432\) \(1.1226\) \(\Gamma_0(N)\)-optimal*
436800.ge1 436800ge3 \([0, -1, 0, -43233, 3458337]\) \(17454600584/93639\) \(47943168000000\) \([2]\) \(1572864\) \(1.4692\) \(\Gamma_0(N)\)-optimal*
436800.ge4 436800ge4 \([0, -1, 0, 16767, -117663]\) \(1018108216/599781\) \(-307087872000000\) \([2]\) \(1572864\) \(1.4692\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 436800ge1.