Properties

Label 97461.q
Number of curves $6$
Conductor $97461$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 97461.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
97461.q1 97461m6 \([1, -1, 0, -8896743, -10210176684]\) \(908031902324522977/161726530797\) \(13870657209245728437\) \([2]\) \(3145728\) \(2.6774\)  
97461.q2 97461m4 \([1, -1, 0, -612558, -125009865]\) \(296380748763217/92608836489\) \(7942700675984789169\) \([2, 2]\) \(1572864\) \(2.3308\)  
97461.q3 97461m2 \([1, -1, 0, -239913, 43798320]\) \(17806161424897/668584449\) \(57341894751652329\) \([2, 2]\) \(786432\) \(1.9843\)  
97461.q4 97461m1 \([1, -1, 0, -237708, 44667531]\) \(17319700013617/25857\) \(2217654590697\) \([2]\) \(393216\) \(1.6377\) \(\Gamma_0(N)\)-optimal
97461.q5 97461m3 \([1, -1, 0, 97452, 156950541]\) \(1193377118543/124806800313\) \(-10704195137267595873\) \([2]\) \(1572864\) \(2.3308\)  
97461.q6 97461m5 \([1, -1, 0, 1709307, -848038626]\) \(6439735268725823/7345472585373\) \(-629992690559283548133\) \([2]\) \(3145728\) \(2.6774\)  

Rank

sage: E.rank()
 

The elliptic curves in class 97461.q have rank \(0\).

Complex multiplication

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

Modular form 97461.2.a.q

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.