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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 12936j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12936.t3 | 12936j1 | \([0, 1, 0, -604, 5312]\) | \(810448/33\) | \(993898752\) | \([2]\) | \(6144\) | \(0.49222\) | \(\Gamma_0(N)\)-optimal |
12936.t2 | 12936j2 | \([0, 1, 0, -1584, -17424]\) | \(3650692/1089\) | \(131194635264\) | \([2, 2]\) | \(12288\) | \(0.83879\) | |
12936.t1 | 12936j3 | \([0, 1, 0, -23144, -1362768]\) | \(5690357426/891\) | \(214682130432\) | \([2]\) | \(24576\) | \(1.1854\) | |
12936.t4 | 12936j4 | \([0, 1, 0, 4296, -111504]\) | \(36382894/43923\) | \(-10583033911296\) | \([2]\) | \(24576\) | \(1.1854\) |
Rank
sage: E.rank()
The elliptic curves in class 12936j have rank \(0\).
Complex multiplication
The elliptic curves in class 12936j do not have complex multiplication.Modular form 12936.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.