Properties

Label 235200gk
Number of curves $8$
Conductor $235200$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("235200.gk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 235200gk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
235200.gk7 235200gk1 [0, -1, 0, -3216033, 782171937] [2] 10616832 \(\Gamma_0(N)\)-optimal
235200.gk5 235200gk2 [0, -1, 0, -28304033, -57396900063] [2, 2] 21233664  
235200.gk4 235200gk3 [0, -1, 0, -210192033, 1173000155937] [2] 31850496  
235200.gk6 235200gk4 [0, -1, 0, -6352033, -144173156063] [2] 42467328  
235200.gk2 235200gk5 [0, -1, 0, -451664033, -3694482660063] [2] 42467328  
235200.gk3 235200gk6 [0, -1, 0, -211760033, 1154612219937] [2, 2] 63700992  
235200.gk8 235200gk7 [0, -1, 0, 57151967, 3886489227937] [2] 127401984  
235200.gk1 235200gk8 [0, -1, 0, -505760033, -2754117780063] [2] 127401984  

Rank

sage: E.rank()
 

The elliptic curves in class 235200gk have rank \(0\).

Modular form 235200.2.a.gk

sage: E.q_eigenform(10)
 
\( q - q^{3} + q^{9} - 2q^{13} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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.