Properties

Label 32851k
Number of curves $3$
Conductor $32851$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 32851k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32851.f2 32851k1 \([0, -1, 1, -2647, -51640]\) \(-43614208/91\) \(-4281175171\) \([]\) \(25272\) \(0.73371\) \(\Gamma_0(N)\)-optimal
32851.f3 32851k2 \([0, -1, 1, 4573, -262103]\) \(224755712/753571\) \(-35452411591051\) \([]\) \(75816\) \(1.2830\)  
32851.f1 32851k3 \([0, -1, 1, -42357, 8283850]\) \(-178643795968/524596891\) \(-24680122906955971\) \([]\) \(227448\) \(1.8323\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32851k have rank \(1\).

Complex multiplication

The elliptic curves in class 32851k do not have complex multiplication.

Modular form 32851.2.a.k

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} - 2 q^{4} - 3 q^{5} + q^{7} + q^{9} - 4 q^{12} - q^{13} - 6 q^{15} + 4 q^{16} - 6 q^{17} + O(q^{20})\) Copy content 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.