Properties

Label 3249c
Number of curves $3$
Conductor $3249$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3249c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3249.d3 3249c1 \([0, 0, 1, 2166, -1715]\) \(32768/19\) \(-651632497731\) \([]\) \(2880\) \(0.95635\) \(\Gamma_0(N)\)-optimal
3249.d2 3249c2 \([0, 0, 1, -30324, -2162300]\) \(-89915392/6859\) \(-235239331680891\) \([]\) \(8640\) \(1.5057\)  
3249.d1 3249c3 \([0, 0, 1, -2499564, -1521053555]\) \(-50357871050752/19\) \(-651632497731\) \([]\) \(25920\) \(2.0550\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3249c have rank \(1\).

Complex multiplication

The elliptic curves in class 3249c do not have complex multiplication.

Modular form 3249.2.a.c

sage: E.q_eigenform(10)
 
\(q - 2q^{4} - 3q^{5} - q^{7} - 3q^{11} + 4q^{13} + 4q^{16} + 3q^{17} + O(q^{20})\)  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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.