Properties

Label 414050.eq
Number of curves $2$
Conductor $414050$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("eq1")
 
E.isogeny_class()
 

Elliptic curves in class 414050.eq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414050.eq1 414050eq2 \([1, 0, 0, -93828043, 349814173617]\) \(-6434774386429585/140608\) \(-1996173994774523200\) \([]\) \(42456960\) \(3.0378\) \(\Gamma_0(N)\)-optimal*
414050.eq2 414050eq1 \([1, 0, 0, -1080843, 546767857]\) \(-9836106385/3407872\) \(-48380643092286668800\) \([]\) \(14152320\) \(2.4885\) \(\Gamma_0(N)\)-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 414050.eq1.

Rank

sage: E.rank()
 

The elliptic curves in class 414050.eq have rank \(1\).

Complex multiplication

The elliptic curves in class 414050.eq do not have complex multiplication.

Modular form 414050.2.a.eq

sage: E.q_eigenform(10)
 
\(q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} + q^{8} + q^{9} + 3 q^{11} - 2 q^{12} + q^{16} + 3 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.