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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 439569z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
439569.z4 | 439569z1 | \([1, -1, 1, -236936849, -1403712409264]\) | \(17319700013617/25857\) | \(2196136377829484881713\) | \([2]\) | \(49545216\) | \(3.3638\) | \(\Gamma_0(N)\)-optimal* |
439569.z3 | 439569z2 | \([1, -1, 1, -239134694, -1376341326772]\) | \(17806161424897/668584449\) | \(56785498321536990586453041\) | \([2, 2]\) | \(99090432\) | \(3.7104\) | \(\Gamma_0(N)\)-optimal* |
439569.z2 | 439569z3 | \([1, -1, 1, -610570499, 3938905042778]\) | \(296380748763217/92608836489\) | \(7865631539697392974191001401\) | \([2, 2]\) | \(198180864\) | \(4.0570\) | \(\Gamma_0(N)\)-optimal* |
439569.z5 | 439569z4 | \([1, -1, 1, 97135591, -4939864790974]\) | \(1193377118543/124806800313\) | \(-10600330833734758092416243817\) | \([2]\) | \(198180864\) | \(4.0570\) | |
439569.z1 | 439569z5 | \([1, -1, 1, -8867874164, 321376081297106]\) | \(908031902324522977/161726530797\) | \(13736068281063240698341006173\) | \([2]\) | \(396361728\) | \(4.4035\) | \(\Gamma_0(N)\)-optimal* |
439569.z6 | 439569z6 | \([1, -1, 1, 1703760286, 26673039209990]\) | \(6439735268725823/7345472585373\) | \(-623879783311553616901620729357\) | \([2]\) | \(396361728\) | \(4.4035\) |
Rank
sage: E.rank()
The elliptic curves in class 439569z have rank \(2\).
Complex multiplication
The elliptic curves in class 439569z do not have complex multiplication.Modular form 439569.2.a.z
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.