Properties

Label 463680.dk
Number of curves $4$
Conductor $463680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dk1")
 
E.isogeny_class()
 

Elliptic curves in class 463680.dk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.dk1 463680dk4 \([0, 0, 0, -2909388, -1910045072]\) \(14251520160844849/264449745\) \(50537133271941120\) \([2]\) \(7864320\) \(2.3301\)  
463680.dk2 463680dk2 \([0, 0, 0, -187788, -27786512]\) \(3832302404449/472410225\) \(90278999890329600\) \([2, 2]\) \(3932160\) \(1.9835\)  
463680.dk3 463680dk1 \([0, 0, 0, -46668, 3429232]\) \(58818484369/7455105\) \(1424692751892480\) \([2]\) \(1966080\) \(1.6369\) \(\Gamma_0(N)\)-optimal*
463680.dk4 463680dk3 \([0, 0, 0, 275892, -143335568]\) \(12152722588271/53476250625\) \(-10219470639759360000\) \([2]\) \(7864320\) \(2.3301\)  
*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.dk1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680.dk have rank \(1\).

Complex multiplication

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

Modular form 463680.2.a.dk

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} + 4 q^{11} + 6 q^{13} + 2 q^{17} + O(q^{20})\) 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.