Properties

Label 439569be
Number of curves $4$
Conductor $439569$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("be1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 439569be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
439569.be3 439569be1 [1, -1, 1, -302204, -27246410] [2] 5308416 \(\Gamma_0(N)\)-optimal*
439569.be2 439569be2 [1, -1, 1, -2500049, 1503332848] [2, 2] 10616832 \(\Gamma_0(N)\)-optimal*
439569.be1 439569be3 [1, -1, 1, -39863414, 96884531020] [2] 21233664 \(\Gamma_0(N)\)-optimal*
439569.be4 439569be4 [1, -1, 1, -302204, 4052833048] [2] 21233664  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 439569be1.

Rank

sage: E.rank()
 

The elliptic curves in class 439569be have rank \(0\).

Complex multiplication

The elliptic curves in class 439569be do not have complex multiplication.

Modular form 439569.2.a.be

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + 3q^{8} - 2q^{10} - 4q^{14} - q^{16} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.