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Properties

Label 465690.ce
Number of curves $4$
Conductor $465690$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ce1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 465690.ce have rank \(1\).

Complex multiplication

The elliptic curves in class 465690.ce do not have complex multiplication.

Modular form 465690.2.a.ce

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9} + q^{10} - 6 q^{11} - q^{12} - 2 q^{13} + 2 q^{14} - q^{15} + q^{16} + q^{18} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 465690.ce

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
465690.ce1 465690ce4 \([1, 1, 1, -1870890, 849118287]\) \(15393836938735081/2275690697640\) \(107061873753978420840\) \([2]\) \(20528640\) \(2.5680\)  
465690.ce2 465690ce3 \([1, 1, 1, -1798690, 927729647]\) \(13679527032530281/381633600\) \(17954288931201600\) \([2]\) \(10264320\) \(2.2214\) \(\Gamma_0(N)\)-optimal*
465690.ce3 465690ce2 \([1, 1, 1, -490065, -132112203]\) \(276670733768281/336980250\) \(15853532740850250\) \([2]\) \(6842880\) \(2.0187\)  
465690.ce4 465690ce1 \([1, 1, 1, -38815, -888703]\) \(137467988281/72562500\) \(3413766740062500\) \([2]\) \(3421440\) \(1.6721\) \(\Gamma_0(N)\)-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 465690.ce1.