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Properties

Label 405600.ej
Number of curves $2$
Conductor $405600$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ej1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 405600.ej have rank \(1\).

Complex multiplication

The elliptic curves in class 405600.ej do not have complex multiplication.

Modular form 405600.2.a.ej

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{7} + q^{9} - 2 q^{17} - 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 405600.ej

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.ej1 405600ej2 \([0, 1, 0, -140833, -7207537]\) \(1000000/507\) \(156620298432000000\) \([2]\) \(3096576\) \(1.9919\)  
405600.ej2 405600ej1 \([0, 1, 0, -77458, 8192588]\) \(10648000/117\) \(564736653000000\) \([2]\) \(1548288\) \(1.6453\) \(\Gamma_0(N)\)-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 405600.ej1.