Properties

Label 97461.w
Number of curves $2$
Conductor $97461$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 97461.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97461.w1 97461w2 \([1, -1, 0, -5167500, 4522649543]\) \(177930109857804849/634933\) \(54455740504893\) \([2]\) \(2764800\) \(2.2774\)  
97461.w2 97461w1 \([1, -1, 0, -323115, 70659728]\) \(43499078731809/82055753\) \(7037603640544113\) \([2]\) \(1382400\) \(1.9309\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 97461.w have rank \(1\).

Complex multiplication

The elliptic curves in class 97461.w do not have complex multiplication.

Modular form 97461.2.a.w

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + 4q^{5} - 3q^{8} + 4q^{10} - 6q^{11} + q^{13} - q^{16} + q^{17} - 8q^{19} + O(q^{20})\)  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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.