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("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 439569.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
439569.z1 439569z5 \([1, -1, 1, -8867874164, 321376081297106]\) \(908031902324522977/161726530797\) \(13736068281063240698341006173\) \([2]\) \(396361728\) \(4.4035\) \(\Gamma_0(N)\)-optimal*
439569.z2 439569z3 \([1, -1, 1, -610570499, 3938905042778]\) \(296380748763217/92608836489\) \(7865631539697392974191001401\) \([2, 2]\) \(198180864\) \(4.0570\) \(\Gamma_0(N)\)-optimal*
439569.z3 439569z2 \([1, -1, 1, -239134694, -1376341326772]\) \(17806161424897/668584449\) \(56785498321536990586453041\) \([2, 2]\) \(99090432\) \(3.7104\) \(\Gamma_0(N)\)-optimal*
439569.z4 439569z1 \([1, -1, 1, -236936849, -1403712409264]\) \(17319700013617/25857\) \(2196136377829484881713\) \([2]\) \(49545216\) \(3.3638\) \(\Gamma_0(N)\)-optimal*
439569.z5 439569z4 \([1, -1, 1, 97135591, -4939864790974]\) \(1193377118543/124806800313\) \(-10600330833734758092416243817\) \([2]\) \(198180864\) \(4.0570\)  
439569.z6 439569z6 \([1, -1, 1, 1703760286, 26673039209990]\) \(6439735268725823/7345472585373\) \(-623879783311553616901620729357\) \([2]\) \(396361728\) \(4.4035\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 439569.z1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 439569.2.a.z

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.