Properties

Label 3696w
Number of curves 6
Conductor 3696
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("3696.t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3696w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3696.t4 3696w1 [0, 1, 0, -544, -5068] [2] 1280 \(\Gamma_0(N)\)-optimal
3696.t3 3696w2 [0, 1, 0, -624, -3564] [2, 2] 2560  
3696.t2 3696w3 [0, 1, 0, -4544, 114036] [2, 4] 5120  
3696.t6 3696w4 [0, 1, 0, 2016, -23628] [2] 5120  
3696.t1 3696w5 [0, 1, 0, -72304, 7459220] [4] 10240  
3696.t5 3696w6 [0, 1, 0, 496, 357972] [4] 10240  

Rank

sage: E.rank()
 

The elliptic curves in class 3696w have rank \(0\).

Modular form 3696.2.a.t

sage: E.q_eigenform(10)
 
\( q + q^{3} - 2q^{5} - q^{7} + q^{9} + q^{11} + 6q^{13} - 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.