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Properties

Label 9251.d
Number of curves $3$
Conductor $9251$
CM no
Rank $1$
Graph

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

Elliptic curves in class 9251.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9251.d1 9251a3 \([0, 1, 1, -6576900, -6494214623]\) \(-52893159101157376/11\) \(-6543056531\) \([]\) \(126000\) \(2.1804\)  
9251.d2 9251a2 \([0, 1, 1, -8690, -567683]\) \(-122023936/161051\) \(-95796890670371\) \([]\) \(25200\) \(1.3756\)  
9251.d3 9251a1 \([0, 1, 1, -280, 4197]\) \(-4096/11\) \(-6543056531\) \([]\) \(5040\) \(0.57092\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 9251.d have rank \(1\).

Complex multiplication

The elliptic curves in class 9251.d do not have complex multiplication.

Modular form 9251.2.a.d

sage: E.q_eigenform(10)
 
\(q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 4 q^{13} - 4 q^{14} + q^{15} - 4 q^{16} + 2 q^{17} - 4 q^{18} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.