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Properties

Label 436800.po
Number of curves $4$
Conductor $436800$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("po1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 436800.po have rank \(1\).

Complex multiplication

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

Modular form 436800.2.a.po

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{7} + q^{9} - 4 q^{11} - q^{13} - 6 q^{17} - 8 q^{19} + O(q^{20})\) 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 436800.po

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
436800.po1 436800po4 \([0, 1, 0, -6989633, -7114951137]\) \(9219915604149769/511875\) \(2096640000000000\) \([2]\) \(12582912\) \(2.4079\)  
436800.po2 436800po2 \([0, 1, 0, -437633, -110863137]\) \(2263054145689/16769025\) \(68685926400000000\) \([2, 2]\) \(6291456\) \(2.0613\)  
436800.po3 436800po3 \([0, 1, 0, -157633, -250583137]\) \(-105756712489/6558605235\) \(-26864047042560000000\) \([2]\) \(12582912\) \(2.4079\)  
436800.po4 436800po1 \([0, 1, 0, -45633, 856863]\) \(2565726409/1404585\) \(5753180160000000\) \([2]\) \(3145728\) \(1.7148\) \(\Gamma_0(N)\)-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 436800.po1.