Properties

Label 463680.k
Number of curves $4$
Conductor $463680$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 463680.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.k1 463680k3 \([0, 0, 0, -26496588, 52496825488]\) \(10765299591712341649/20708625\) \(3957479866368000\) \([2]\) \(17301504\) \(2.6745\) \(\Gamma_0(N)\)-optimal*
463680.k2 463680k2 \([0, 0, 0, -1656588, 819689488]\) \(2630872462131649/3645140625\) \(696597221376000000\) \([2, 2]\) \(8650752\) \(2.3279\) \(\Gamma_0(N)\)-optimal*
463680.k3 463680k4 \([0, 0, 0, -1192908, 1288377232]\) \(-982374577874929/3183837890625\) \(-608440896000000000000\) \([2]\) \(17301504\) \(2.6745\)  
463680.k4 463680k1 \([0, 0, 0, -133068, 4910992]\) \(1363569097969/734582625\) \(140380925755392000\) \([2]\) \(4325376\) \(1.9814\) \(\Gamma_0(N)\)-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 463680.k1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680.k have rank \(2\).

Complex multiplication

The elliptic curves in class 463680.k do not have complex multiplication.

Modular form 463680.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} - 4 q^{11} - 2 q^{13} + 2 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.