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Properties

Label 477950dd
Number of curves $2$
Conductor $477950$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 477950dd have rank \(0\).

Complex multiplication

The elliptic curves in class 477950dd do not have complex multiplication.

Modular form 477950.2.a.dd

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + 3 q^{7} + q^{8} - 2 q^{9} + q^{12} - q^{13} + 3 q^{14} + q^{16} - 2 q^{17} - 2 q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 477950dd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
477950.dd2 477950dd1 \([1, 0, 0, -1270563, -546512383]\) \(8194759433281/82837504\) \(2292995178496000000\) \([]\) \(9408000\) \(2.3411\) \(\Gamma_0(N)\)-optimal*
477950.dd1 477950dd2 \([1, 0, 0, -70724563, 228917645617]\) \(1413378216646643521/49232902384\) \(1362798277817209750000\) \([]\) \(47040000\) \(3.1458\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 477950dd1.