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Properties

Label 485184.em
Number of curves $2$
Conductor $485184$
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, -153692621, 733428812253]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -153692621, 733428812253]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -153692621, 733428812253]) 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 485184.em have rank \(0\).

Complex multiplication

The elliptic curves in class 485184.em do not have complex multiplication.

Modular form 485184.2.a.em

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);
 
\(q - q^{3} + 4 q^{5} - q^{7} + q^{9} - 2 q^{13} - 4 q^{15} + 4 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 485184.em

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
485184.em1 485184em1 \([0, -1, 0, -153692621, 733428812253]\) \(8334147900493981696/232793757\) \(11214835086709675008\) \([2]\) \(66355200\) \(3.1662\) \(\Gamma_0(N)\)-optimal
485184.em2 485184em2 \([0, -1, 0, -153497681, 735381916113]\) \(-518904725785387216/2753286252003\) \(-2122232256441043347750912\) \([2]\) \(132710400\) \(3.5128\)