Properties

Label 439569.z
Number of curves $6$
Conductor $439569$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("439569.z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 439569.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
439569.z1 439569z5 [1, -1, 1, -8867874164, 321376081297106] [2] 396361728 \(\Gamma_0(N)\)-optimal*
439569.z2 439569z3 [1, -1, 1, -610570499, 3938905042778] [2, 2] 198180864 \(\Gamma_0(N)\)-optimal*
439569.z3 439569z2 [1, -1, 1, -239134694, -1376341326772] [2, 2] 99090432 \(\Gamma_0(N)\)-optimal*
439569.z4 439569z1 [1, -1, 1, -236936849, -1403712409264] [2] 49545216 \(\Gamma_0(N)\)-optimal*
439569.z5 439569z4 [1, -1, 1, 97135591, -4939864790974] [2] 198180864  
439569.z6 439569z6 [1, -1, 1, 1703760286, 26673039209990] [2] 396361728  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 439569.z4.

Rank

sage: E.rank()
 

The elliptic curves in class 439569.z have rank \(2\).

Modular form 439569.2.a.z

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + 2q^{5} + 3q^{8} - 2q^{10} - 4q^{11} - q^{16} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.