Properties

Label 40656.bj
Number of curves $4$
Conductor $40656$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bj1")
 
E.isogeny_class()
 

Elliptic curves in class 40656.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40656.bj1 40656bj4 \([0, -1, 0, -77918232, 264758160240]\) \(7209828390823479793/49509306\) \(359255063128743936\) \([2]\) \(2211840\) \(2.9683\)  
40656.bj2 40656bj3 \([0, -1, 0, -6789592, 581450608]\) \(4770223741048753/2740574865798\) \(19886471372095367897088\) \([2]\) \(2211840\) \(2.9683\)  
40656.bj3 40656bj2 \([0, -1, 0, -4872952, 4132601200]\) \(1763535241378513/4612311396\) \(33468379090981502976\) \([2, 2]\) \(1105920\) \(2.6217\)  
40656.bj4 40656bj1 \([0, -1, 0, -187832, 114642288]\) \(-100999381393/723148272\) \(-5247390826056056832\) \([2]\) \(552960\) \(2.2752\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 40656.bj have rank \(1\).

Complex multiplication

The elliptic curves in class 40656.bj do not have complex multiplication.

Modular form 40656.2.a.bj

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - q^{7} + q^{9} + 2 q^{13} - 2 q^{15} + 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.