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Properties

Label 402930.ek
Number of curves $2$
Conductor $402930$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ek1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 402930.ek have rank \(0\).

Complex multiplication

The elliptic curves in class 402930.ek do not have complex multiplication.

Modular form 402930.2.a.ek

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - 2 q^{13} + q^{14} + q^{16} + 3 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 402930.ek

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
402930.ek1 402930ek2 \([1, -1, 1, -8019435227, -276414394165621]\) \(-44164307457093068844199489/1823508000000000\) \(-2355002173215252000000000\) \([]\) \(256608000\) \(4.1630\)  
402930.ek2 402930ek1 \([1, -1, 1, -90818267, -444448782709]\) \(-64144540676215729729/28962038218752000\) \(-37403544676472023154688000\) \([]\) \(85536000\) \(3.6137\) \(\Gamma_0(N)\)-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 402930.ek1.