Properties

Label 109200fz
Number of curves 8
Conductor 109200
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("109200.ge1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 109200fz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
109200.ge7 109200fz1 [0, 1, 0, -10290008, -12640236012] [2] 5308416 \(\Gamma_0(N)\)-optimal
109200.ge6 109200fz2 [0, 1, 0, -16562008, 4582675988] [2, 2] 10616832  
109200.ge5 109200fz3 [0, 1, 0, -63570008, 186566483988] [2] 15925248  
109200.ge8 109200fz4 [0, 1, 0, 65085992, 36425395988] [2] 21233664  
109200.ge4 109200fz5 [0, 1, 0, -198562008, 1075106675988] [2] 21233664  
109200.ge2 109200fz6 [0, 1, 0, -1004762008, 12258295075988] [2, 2] 31850496  
109200.ge3 109200fz7 [0, 1, 0, -992414008, 12574280395988] [2] 63700992  
109200.ge1 109200fz8 [0, 1, 0, -16076182008, 784547998715988] [2] 63700992  

Rank

sage: E.rank()
 

The elliptic curves in class 109200fz have rank \(0\).

Modular form 109200.2.a.ge

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

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.