Properties

Label 3696.s
Number of curves $4$
Conductor $3696$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3696.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3696.s1 3696u4 \([0, 1, 0, -3624, 82740]\) \(1285429208617/614922\) \(2518720512\) \([2]\) \(3072\) \(0.75881\)  
3696.s2 3696u3 \([0, 1, 0, -2024, -35148]\) \(223980311017/4278582\) \(17525071872\) \([2]\) \(3072\) \(0.75881\)  
3696.s3 3696u2 \([0, 1, 0, -264, 756]\) \(498677257/213444\) \(874266624\) \([2, 2]\) \(1536\) \(0.41223\)  
3696.s4 3696u1 \([0, 1, 0, 56, 116]\) \(4657463/3696\) \(-15138816\) \([2]\) \(768\) \(0.065661\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3696.s have rank \(1\).

Complex multiplication

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

Modular form 3696.2.a.s

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.