Properties

Label 463680.kj
Number of curves $4$
Conductor $463680$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("kj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 463680.kj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.kj1 463680kj4 \([0, 0, 0, -6677292, -6641232176]\) \(344577854816148242/2716875\) \(259601448960000\) \([2]\) \(7864320\) \(2.3568\)  
463680.kj2 463680kj2 \([0, 0, 0, -417612, -103622384]\) \(168591300897604/472410225\) \(22569749972582400\) \([2, 2]\) \(3932160\) \(2.0102\)  
463680.kj3 463680kj3 \([0, 0, 0, -252012, -186621104]\) \(-18524646126002/146738831715\) \(-14021113717749841920\) \([2]\) \(7864320\) \(2.3568\)  
463680.kj4 463680kj1 \([0, 0, 0, -36732, -175376]\) \(458891455696/264449745\) \(3158570829496320\) \([2]\) \(1966080\) \(1.6636\) \(\Gamma_0(N)\)-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 463680.kj1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680.kj have rank \(0\).

Complex multiplication

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

Modular form 463680.2.a.kj

sage: E.q_eigenform(10)
 
\(q + q^{5} - q^{7} + 4 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\)  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.