Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -93175008, -346008031488]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -93175008, -346008031488]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -93175008, -346008031488]) 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 418800.c have rank \(1\).

Complex multiplication

The elliptic curves in class 418800.c do not have complex multiplication.

Modular form 418800.2.a.c

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);
 
\(q - q^{3} - 4 q^{7} + q^{9} + 4 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 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 labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 418800.c

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
418800.c1 418800c4 \([0, -1, 0, -93175008, -346008031488]\) \(1397790417785316543001/640892891563200\) \(41017145060044800000000\) \([2]\) \(71663616\) \(3.2960\)  
418800.c2 418800c3 \([0, -1, 0, -51447008, 139610656512]\) \(235301185027835613721/4636822725000000\) \(296756654400000000000000\) \([2]\) \(71663616\) \(3.2960\) \(\Gamma_0(N)\)-optimal*
418800.c3 418800c2 \([0, -1, 0, -6775008, -3518431488]\) \(537369779909439001/227309898240000\) \(14547833487360000000000\) \([2, 2]\) \(35831808\) \(2.9494\) \(\Gamma_0(N)\)-optimal*
418800.c4 418800c1 \([0, -1, 0, 1416992, -405471488]\) \(4916382075769319/3952292659200\) \(-252946730188800000000\) \([2]\) \(17915904\) \(2.6028\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 418800.c1.