Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("ca1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 9702ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9702.bu3 9702ca1 \([1, -1, 1, -2435, 44223]\) \(18609625/1188\) \(101890151748\) \([2]\) \(11520\) \(0.86288\) \(\Gamma_0(N)\)-optimal
9702.bu4 9702ca2 \([1, -1, 1, 1975, 183579]\) \(9938375/176418\) \(-15130687534578\) \([2]\) \(23040\) \(1.2095\)  
9702.bu1 9702ca3 \([1, -1, 1, -35510, -2556795]\) \(57736239625/255552\) \(21917703753792\) \([2]\) \(34560\) \(1.4122\)  
9702.bu2 9702ca4 \([1, -1, 1, -17870, -5111067]\) \(-7357983625/127552392\) \(-10939673886111432\) \([2]\) \(69120\) \(1.7588\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 9702.2.a.ca

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.