Properties

Label 38640.bx
Number of curves $4$
Conductor $38640$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bx1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 38640.bx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.bx1 38640q4 \([0, 1, 0, -131696, 7482804]\) \(123343086124179938/59429226844575\) \(121711056577689600\) \([2]\) \(393216\) \(1.9721\)  
38640.bx2 38640q2 \([0, 1, 0, -108696, 13748004]\) \(138697437757771876/106292300625\) \(108843315840000\) \([2, 2]\) \(196608\) \(1.6255\)  
38640.bx3 38640q1 \([0, 1, 0, -108676, 13753340]\) \(554483565352358224/326025\) \(83462400\) \([2]\) \(98304\) \(1.2789\) \(\Gamma_0(N)\)-optimal
38640.bx4 38640q3 \([0, 1, 0, -86016, 19672020]\) \(-34366597532983298/61980408984375\) \(-126935877600000000\) \([2]\) \(393216\) \(1.9721\)  

Rank

sage: E.rank()
 

The elliptic curves in class 38640.bx have rank \(1\).

Complex multiplication

The elliptic curves in class 38640.bx do not have complex multiplication.

Modular form 38640.2.a.bx

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} - 6 q^{13} - q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.