Properties

Label 405600.q
Number of curves $2$
Conductor $405600$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 405600.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.q1 405600q2 \([0, -1, 0, -2435008, -1192138988]\) \(18821096/3645\) \(309227201716680000000\) \([2]\) \(20127744\) \(2.6487\) \(\Gamma_0(N)\)-optimal*
405600.q2 405600q1 \([0, -1, 0, 311242, -110116488]\) \(314432/675\) \(-7158037076775000000\) \([2]\) \(10063872\) \(2.3022\) \(\Gamma_0(N)\)-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 405600.q1.

Rank

sage: E.rank()
 

The elliptic curves in class 405600.q have rank \(1\).

Complex multiplication

The elliptic curves in class 405600.q do not have complex multiplication.

Modular form 405600.2.a.q

sage: E.q_eigenform(10)
 
\(q - q^{3} - 4q^{7} + q^{9} + 4q^{11} + 8q^{19} + O(q^{20})\)  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.