Properties

Label 3211a
Number of curves $3$
Conductor $3211$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3211a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3211.a3 3211a1 \([0, 1, 1, 113, 17]\) \(32768/19\) \(-91709371\) \([]\) \(720\) \(0.21730\) \(\Gamma_0(N)\)-optimal
3211.a2 3211a2 \([0, 1, 1, -1577, -26178]\) \(-89915392/6859\) \(-33107082931\) \([]\) \(2160\) \(0.76661\)  
3211.a1 3211a3 \([0, 1, 1, -130017, -18088053]\) \(-50357871050752/19\) \(-91709371\) \([]\) \(6480\) \(1.3159\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3211a have rank \(1\).

Complex multiplication

The elliptic curves in class 3211a do not have complex multiplication.

Modular form 3211.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2q^{3} - 2q^{4} - 3q^{5} + q^{7} + q^{9} - 3q^{11} + 4q^{12} + 6q^{15} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20})\)  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 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.