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Properties

Label 9690j
Number of curves $4$
Conductor $9690$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -1429, 26336]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -1429, 26336]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -1429, 26336]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 9690j have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(17\)\(1 + T\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 13 T^{2}\) 1.13.a
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 9690j do not have complex multiplication.

Modular form 9690.2.a.j

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} + q^{12} + 6 q^{13} - q^{15} + q^{16} - q^{17} - q^{18} - q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 9690j

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9690.k4 9690j1 \([1, 0, 1, -1429, 26336]\) \(-322391399464009/115780151040\) \(-115780151040\) \([2]\) \(12288\) \(0.83362\) \(\Gamma_0(N)\)-optimal
9690.k3 9690j2 \([1, 0, 1, -24549, 1478272]\) \(1636061778667305289/135585968400\) \(135585968400\) \([2, 2]\) \(24576\) \(1.1802\)  
9690.k2 9690j3 \([1, 0, 1, -26249, 1261352]\) \(2000037860254622089/467727326149140\) \(467727326149140\) \([2]\) \(49152\) \(1.5268\)  
9690.k1 9690j4 \([1, 0, 1, -392769, 94711576]\) \(6700909177116065071369/46027500\) \(46027500\) \([2]\) \(49152\) \(1.5268\)