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Properties

Label 450528f
Number of curves $4$
Conductor $450528$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 450528f have rank \(1\).

Complex multiplication

The elliptic curves in class 450528f do not have complex multiplication.

Modular form 450528.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - q^{13} + 2 q^{15} - 6 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 450528f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450528.f3 450528f1 \([0, -1, 0, -84594, -9396576]\) \(22235451328/123201\) \(370950373445184\) \([2, 2]\) \(2322432\) \(1.6384\) \(\Gamma_0(N)\)-optimal*
450528.f2 450528f2 \([0, -1, 0, -133329, 2718945]\) \(1360251712/771147\) \(148599971821596672\) \([2]\) \(4644864\) \(1.9849\) \(\Gamma_0(N)\)-optimal*
450528.f4 450528f3 \([0, -1, 0, -37664, -19815036]\) \(-245314376/6908733\) \(-166414044456331776\) \([2]\) \(4644864\) \(1.9849\)  
450528.f1 450528f4 \([0, -1, 0, -1351704, -604431432]\) \(11339065490696/351\) \(8454709366272\) \([2]\) \(4644864\) \(1.9849\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 450528f1.