Properties

Label 428400ek
Number of curves $4$
Conductor $428400$
CM no
Rank $0$
Graph

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

Elliptic curves in class 428400ek

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
428400.ek4 428400ek1 \([0, 0, 0, 3525, 21766250]\) \(103823/4386816\) \(-204671287296000000\) \([2]\) \(4718592\) \(2.0006\) \(\Gamma_0(N)\)-optimal*
428400.ek3 428400ek2 \([0, 0, 0, -1148475, 465286250]\) \(3590714269297/73410624\) \(3425046073344000000\) \([2, 2]\) \(9437184\) \(2.3471\) \(\Gamma_0(N)\)-optimal*
428400.ek1 428400ek3 \([0, 0, 0, -18284475, 30093430250]\) \(14489843500598257/6246072\) \(291416735232000000\) \([2]\) \(18874368\) \(2.6937\) \(\Gamma_0(N)\)-optimal*
428400.ek2 428400ek4 \([0, 0, 0, -2444475, -777577750]\) \(34623662831857/14438442312\) \(673639964508672000000\) \([2]\) \(18874368\) \(2.6937\)  
*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 428400ek1.

Rank

sage: E.rank()
 

The elliptic curves in class 428400ek have rank \(0\).

Complex multiplication

The elliptic curves in class 428400ek do not have complex multiplication.

Modular form 428400.2.a.ek

sage: E.q_eigenform(10)
 
\(q - q^{7} + 6q^{13} + q^{17} + O(q^{20})\)  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 Cremona 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 Cremona labels.