Properties

Label 3136s
Number of curves $4$
Conductor $3136$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3136s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3136.p4 3136s1 [0, 0, 0, 196, 5488] [2] 1536 \(\Gamma_0(N)\)-optimal
3136.p3 3136s2 [0, 0, 0, -3724, 82320] [2, 2] 3072  
3136.p2 3136s3 [0, 0, 0, -11564, -378672] [2] 6144  
3136.p1 3136s4 [0, 0, 0, -58604, 5460560] [2] 6144  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 3136s do not have complex multiplication.

Modular form 3136.2.a.s

sage: E.q_eigenform(10)
 
\( q + 2q^{5} - 3q^{9} - 4q^{11} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.