Properties

Label 235200ux
Number of curves 8
Conductor 235200
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("235200.ux1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 235200ux

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
235200.ux7 235200ux1 [0, 1, 0, -39005633, 93715660863] [2] 21233664 \(\Gamma_0(N)\)-optimal
235200.ux6 235200ux2 [0, 1, 0, -45277633, 61546572863] [2, 2] 42467328  
235200.ux5 235200ux3 [0, 1, 0, -115445633, -362883979137] [2] 63700992  
235200.ux8 235200ux4 [0, 1, 0, 150722367, 452174572863] [2] 84934656  
235200.ux4 235200ux5 [0, 1, 0, -341629633, -2387802707137] [2] 84934656  
235200.ux2 235200ux6 [0, 1, 0, -1721077633, -27480402827137] [2, 2] 127401984  
235200.ux3 235200ux7 [0, 1, 0, -1595637633, -31655924107137] [2] 254803968  
235200.ux1 235200ux8 [0, 1, 0, -27536629633, -1758800397707137] [2] 254803968  

Rank

sage: E.rank()
 

The elliptic curves in class 235200ux have rank \(2\).

Modular form 235200.2.a.ux

sage: E.q_eigenform(10)
 
\( q + q^{3} + q^{9} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20}) \)

Isogeny matrix

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.