Properties

Label 3136y
Number of curves 6
Conductor 3136
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("3136.z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3136y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3136.z5 3136y1 [0, -1, 0, -1633, 51969] [2] 3072 \(\Gamma_0(N)\)-optimal
3136.z4 3136y2 [0, -1, 0, -32993, 2316161] [2] 6144  
3136.z6 3136y3 [0, -1, 0, 14047, -1080127] [2] 9216  
3136.z3 3136y4 [0, -1, 0, -111393, -11692351] [2] 18432  
3136.z2 3136y5 [0, -1, 0, -534753, -150770815] [2] 27648  
3136.z1 3136y6 [0, -1, 0, -8562913, -9641661567] [2] 55296  

Rank

sage: E.rank()
 

The elliptic curves in class 3136y have rank \(1\).

Modular form 3136.2.a.z

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.