↑

Properties

Label 436800eu
Number of curves $6$
Conductor $436800$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("eu1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 436800eu have rank \(2\).

Complex multiplication

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

Modular form 436800.2.a.eu

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{7} + q^{9} + 4 q^{11} + q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 436800eu

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
436800.eu5 436800eu1 \([0, -1, 0, -73533, 10051437]\) \(-2748251600896/1124136195\) \(-17986179120000000\) \([2]\) \(3145728\) \(1.8274\) \(\Gamma_0(N)\)-optimal*
436800.eu4 436800eu2 \([0, -1, 0, -1274033, 553877937]\) \(893359210685776/91298025\) \(23372294400000000\) \([2, 2]\) \(6291456\) \(2.1740\) \(\Gamma_0(N)\)-optimal*
436800.eu1 436800eu3 \([0, -1, 0, -20384033, 35429627937]\) \(914732517663095044/9555\) \(9784320000000\) \([2]\) \(12582912\) \(2.5206\) \(\Gamma_0(N)\)-optimal*
436800.eu3 436800eu4 \([0, -1, 0, -1372033, 463815937]\) \(278944461825124/70849130625\) \(72549509760000000000\) \([2, 2]\) \(12582912\) \(2.5206\)  
436800.eu6 436800eu5 \([0, -1, 0, 3359967, 2957579937]\) \(2048324060764798/3031899609375\) \(-6209330400000000000000\) \([2]\) \(25165824\) \(2.8671\)  
436800.eu2 436800eu6 \([0, -1, 0, -7672033, -7795484063]\) \(24385137179326562/1284775885575\) \(2631221013657600000000\) \([2]\) \(25165824\) \(2.8671\)  
*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 436800eu1.