Properties

Label 3696i
Number of curves $4$
Conductor $3696$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3696i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3696.ba3 3696i1 \([0, 1, 0, -586972, 172895372]\) \(87364831012240243408/1760913\) \(450793728\) \([2]\) \(23040\) \(1.6440\) \(\Gamma_0(N)\)-optimal
3696.ba2 3696i2 \([0, 1, 0, -586992, 172882980]\) \(21843440425782779332/3100814593569\) \(3175234143814656\) \([2, 2]\) \(46080\) \(1.9906\)  
3696.ba1 3696i3 \([0, 1, 0, -640232, 139597332]\) \(14171198121996897746/4077720290568771\) \(8351171155084843008\) \([2]\) \(92160\) \(2.3371\)  
3696.ba4 3696i4 \([0, 1, 0, -534072, 205375860]\) \(-8226100326647904626/4152140742401883\) \(-8503584240439056384\) \([2]\) \(92160\) \(2.3371\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 3696i do not have complex multiplication.

Modular form 3696.2.a.i

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2q^{5} - q^{7} + q^{9} - q^{11} - 6q^{13} + 2q^{15} - 2q^{17} + 8q^{19} + O(q^{20})\)  Toggle raw display

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.