Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -609, 5247]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -609, 5247]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -609, 5247]) 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 25536bd have rank \(2\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1 - T\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(23\) \( 1 + 2 T + 23 T^{2}\) 1.23.c
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 25536bd do not have complex multiplication.

Modular form 25536.2.a.bd

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);
 
\(q + q^{3} - 2 q^{5} - q^{7} + q^{9} - 6 q^{11} + 2 q^{13} - 2 q^{15} - 8 q^{17} - q^{19} + O(q^{20})\) 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.

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

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 25536bd

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
25536.cb1 25536bd1 \([0, 1, 0, -609, 5247]\) \(381775972/25137\) \(1647378432\) \([2]\) \(18432\) \(0.51814\) \(\Gamma_0(N)\)-optimal
25536.cb2 25536bd2 \([0, 1, 0, 511, 23391]\) \(112363774/1842183\) \(-241458610176\) \([2]\) \(36864\) \(0.86471\)