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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1008j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1008.g4 | 1008j1 | \([0, 0, 0, 60, -61]\) | \(2048000/1323\) | \(-15431472\) | \([2]\) | \(192\) | \(0.068235\) | \(\Gamma_0(N)\)-optimal |
1008.g3 | 1008j2 | \([0, 0, 0, -255, -502]\) | \(9826000/5103\) | \(952342272\) | \([2]\) | \(384\) | \(0.41481\) | |
1008.g2 | 1008j3 | \([0, 0, 0, -1020, -12913]\) | \(-10061824000/352947\) | \(-4116773808\) | \([2]\) | \(576\) | \(0.61754\) | |
1008.g1 | 1008j4 | \([0, 0, 0, -16455, -812446]\) | \(2640279346000/3087\) | \(576108288\) | \([2]\) | \(1152\) | \(0.96412\) |
Rank
sage: E.rank()
The elliptic curves in class 1008j have rank \(1\).
Complex multiplication
The elliptic curves in class 1008j do not have complex multiplication.Modular form 1008.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.