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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 81120d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
81120.e1 | 81120d1 | \([0, -1, 0, -1789766, -919327020]\) | \(2052450196928704/4317958125\) | \(1333885384919880000\) | \([2]\) | \(1548288\) | \(2.3630\) | \(\Gamma_0(N)\)-optimal |
81120.e2 | 81120d2 | \([0, -1, 0, -1173761, -1562313039]\) | \(-9045718037056/48125390625\) | \(-951468312974400000000\) | \([2]\) | \(3096576\) | \(2.7095\) |
Rank
sage: E.rank()
The elliptic curves in class 81120d have rank \(1\).
Complex multiplication
The elliptic curves in class 81120d do not have complex multiplication.Modular form 81120.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.