Properties

Label 9702.r
Number of curves $4$
Conductor $9702$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 9702.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9702.r1 9702p4 \([1, -1, 0, -17748936, 28785485182]\) \(7209828390823479793/49509306\) \(4246221129022026\) \([2]\) \(294912\) \(2.5985\)  
9702.r2 9702p3 \([1, -1, 0, -1546596, 63354514]\) \(4770223741048753/2740574865798\) \(235048475549590029558\) \([2]\) \(294912\) \(2.5985\)  
9702.r3 9702p2 \([1, -1, 0, -1110006, 449387392]\) \(1763535241378513/4612311396\) \(395580057279014916\) \([2, 2]\) \(147456\) \(2.2519\)  
9702.r4 9702p1 \([1, -1, 0, -42786, 12467524]\) \(-100999381393/723148272\) \(-62021622197292912\) \([2]\) \(73728\) \(1.9053\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 9702.r have rank \(1\).

Complex multiplication

The elliptic curves in class 9702.r do not have complex multiplication.

Modular form 9702.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.