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Properties

Label 462550bo
Number of curves $3$
Conductor $462550$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -589138, -174465219]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -589138, -174465219]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -589138, -174465219]) 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 462550bo have rank \(1\).

Complex multiplication

The elliptic curves in class 462550bo do not have complex multiplication.

Modular form 462550.2.a.bo

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);
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9} - q^{11} - q^{12} - 4 q^{13} - 3 q^{14} + q^{16} + 3 q^{17} - 2 q^{18} + 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

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 462550bo

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
462550.bo2 462550bo1 \([1, 1, 1, -589138, -174465219]\) \(-19465109/22\) \(-25558814574218750\) \([]\) \(5880000\) \(2.0621\) \(\Gamma_0(N)\)-optimal*
462550.bo3 462550bo2 \([1, 1, 1, 4141487, 1828166031]\) \(6761990971/5153632\) \(-5987305666898187500000\) \([]\) \(29400000\) \(2.8668\) \(\Gamma_0(N)\)-optimal*
462550.bo1 462550bo3 \([1, 1, 1, -637646638, 6197338503531]\) \(-24680042791780949/369098752\) \(-428805752815616000000000\) \([]\) \(147000000\) \(3.6715\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 462550bo1.