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Properties

Label 436800il
Number of curves $2$
Conductor $436800$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -512608, -141091538]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -512608, -141091538]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -512608, -141091538]) 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 436800il have rank \(0\).

Complex multiplication

The elliptic curves in class 436800il do not have complex multiplication.

Modular form 436800.2.a.il

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);
 
\(q - q^{3} + q^{7} + q^{9} + q^{13} + 4 q^{17} - 4 q^{19} + O(q^{20})\) 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 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 436800il

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
436800.il1 436800il1 \([0, -1, 0, -512608, -141091538]\) \(14896378491692608/138411\) \(138411000000\) \([2]\) \(1966080\) \(1.7183\) \(\Gamma_0(N)\)-optimal
436800.il2 436800il2 \([0, -1, 0, -512233, -141308663]\) \(-232245467895232/709540923\) \(-45410619072000000\) \([2]\) \(3932160\) \(2.0648\)