Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -25381776, -49220665427]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -25381776, -49220665427]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -25381776, -49220665427]) 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 240825.cg have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1\)
\(13\)\(1\)
\(19\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 240825.cg do not have complex multiplication.

Modular form 240825.2.a.cg

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);
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} + 4 q^{7} - 3 q^{8} + q^{9} - 4 q^{11} - q^{12} + 4 q^{14} - q^{16} - 2 q^{17} + q^{18} + q^{19} + O(q^{20})\) 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 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

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

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

Elliptic curves in class 240825.cg

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
240825.cg1 240825cg4 \([1, 0, 1, -25381776, -49220665427]\) \(23977812996389881/146611125\) \(11057248400783203125\) \([2]\) \(15925248\) \(2.8413\)  
240825.cg2 240825cg3 \([1, 0, 1, -5228526, 3726105073]\) \(209595169258201/41748046875\) \(3148591381072998046875\) \([2]\) \(15925248\) \(2.8413\)  
240825.cg3 240825cg2 \([1, 0, 1, -1616151, -738790427]\) \(6189976379881/456890625\) \(34458184074462890625\) \([2, 2]\) \(7962624\) \(2.4948\)  
240825.cg4 240825cg1 \([1, 0, 1, 94974, -50918177]\) \(1256216039/15582375\) \(-1175205435802734375\) \([2]\) \(3981312\) \(2.1482\) \(\Gamma_0(N)\)-optimal