Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 414736.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.g1 414736g2 \([0, 1, 0, -14109664, -19791928668]\) \(1431644/49\) \(10632485543889473395712\) \([2]\) \(23740416\) \(2.9984\)  
414736.g2 414736g1 \([0, 1, 0, -2186004, 816925276]\) \(21296/7\) \(379731626567481192704\) \([2]\) \(11870208\) \(2.6518\) \(\Gamma_0(N)\)-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 414736.g1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 414736.2.a.g

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} - 2 q^{5} + q^{9} + 2 q^{13} + 4 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\)  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 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.