Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -70805, 6740636]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -70805, 6740636]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -70805, 6740636]) 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 453152x have rank \(1\).

Complex multiplication

The elliptic curves in class 453152x do not have complex multiplication.

Modular form 453152.2.a.x

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);
 
\(q - 3 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{19} + O(q^{20})\) 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 labeled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 453152x

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
453152.x1 453152x1 \([0, 0, 0, -70805, 6740636]\) \(216000/17\) \(3089659810545728\) \([2]\) \(1658880\) \(1.7161\) \(\Gamma_0(N)\)-optimal
453152.x2 453152x2 \([0, 0, 0, 70805, 30332862]\) \(27000/289\) \(-420193734234219008\) \([2]\) \(3317760\) \(2.0627\)