Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, 15967, 3140063]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, 15967, 3140063]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, 15967, 3140063]) 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 17600cf have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(5\)\(1\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(13\) \( 1 + 13 T^{2}\) 1.13.a
\(17\) \( 1 + T + 17 T^{2}\) 1.17.b
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - T + 29 T^{2}\) 1.29.ab
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 17600cf do not have complex multiplication.

Modular form 17600.2.a.cf

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);
 
\(q + q^{3} + 3 q^{7} - 2 q^{9} + q^{11} - 6 q^{13} + 7 q^{17} + 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

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 17600cf

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
17600.ck2 17600cf1 \([0, 1, 0, 15967, 3140063]\) \(109902239/1100000\) \(-4505600000000000\) \([]\) \(92160\) \(1.6848\) \(\Gamma_0(N)\)-optimal
17600.ck1 17600cf2 \([0, 1, 0, -9504033, 11274260063]\) \(-23178622194826561/1610510\) \(-6596648960000000\) \([]\) \(460800\) \(2.4895\)