Properties

Label 414736by
Number of curves $2$
Conductor $414736$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("by1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 414736by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.by2 414736by1 \([0, -1, 0, -734428, -311051040]\) \(-9826000/3703\) \(-16510070720325269248\) \([2]\) \(8110080\) \(2.3974\) \(\Gamma_0(N)\)-optimal*
414736.by1 414736by2 \([0, -1, 0, -12658088, -17328498592]\) \(12576878500/1127\) \(20099216529091632128\) \([2]\) \(16220160\) \(2.7439\) \(\Gamma_0(N)\)-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 414736by1.

Rank

sage: E.rank()
 

The elliptic curves in class 414736by have rank \(1\).

Complex multiplication

The elliptic curves in class 414736by do not have complex multiplication.

Modular form 414736.2.a.by

sage: E.q_eigenform(10)
 
\(q + 2q^{3} + q^{9} + 4q^{11} - 6q^{13} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.