Properties

Label 87451d
Number of curves $3$
Conductor $87451$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 87451d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
87451.d2 87451d1 \([0, -1, 1, -7047, -225773]\) \(-43614208/91\) \(-80762834971\) \([]\) \(119880\) \(0.97849\) \(\Gamma_0(N)\)-optimal
87451.d3 87451d2 \([0, -1, 1, 12173, -1135840]\) \(224755712/753571\) \(-668797036394851\) \([]\) \(359640\) \(1.5278\)  
87451.d1 87451d3 \([0, -1, 1, -112757, 35955877]\) \(-178643795968/524596891\) \(-465581671803655771\) \([]\) \(1078920\) \(2.0771\)  

Rank

sage: E.rank()
 

The elliptic curves in class 87451d have rank \(1\).

Complex multiplication

The elliptic curves in class 87451d do not have complex multiplication.

Modular form 87451.2.a.d

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} - 2 q^{4} - 3 q^{5} + q^{7} + q^{9} - 4 q^{12} - q^{13} - 6 q^{15} + 4 q^{16} + 6 q^{17} - 7 q^{19} + O(q^{20})\) 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.