Properties

Label 428400.ey
Number of curves $2$
Conductor $428400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ey1")
 
E.isogeny_class()
 

Elliptic curves in class 428400.ey

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
428400.ey1 428400ey2 \([0, 0, 0, -135660675, -607419130750]\) \(5918043195362419129/8515734343200\) \(397310101516339200000000\) \([2]\) \(70778880\) \(3.4307\) \(\Gamma_0(N)\)-optimal*
428400.ey2 428400ey1 \([0, 0, 0, -6060675, -15017530750]\) \(-527690404915129/1782829440000\) \(-83179690352640000000000\) \([2]\) \(35389440\) \(3.0841\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 428400.ey1.

Rank

sage: E.rank()
 

The elliptic curves in class 428400.ey have rank \(1\).

Complex multiplication

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

Modular form 428400.2.a.ey

sage: E.q_eigenform(10)
 
\(q - q^{7} + 2 q^{11} - 4 q^{13} - q^{17} + 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.