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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 8415l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8415.i1 | 8415l1 | \([0, 0, 1, -118398, -15680691]\) | \(-251784668965666816/353546875\) | \(-257735671875\) | \([]\) | \(35136\) | \(1.4619\) | \(\Gamma_0(N)\)-optimal |
8415.i2 | 8415l2 | \([0, 0, 1, -86898, -24206166]\) | \(-99546392709922816/289614925147075\) | \(-211129280432217675\) | \([3]\) | \(105408\) | \(2.0112\) |
Rank
sage: E.rank()
The elliptic curves in class 8415l have rank \(0\).
Complex multiplication
The elliptic curves in class 8415l do not have complex multiplication.Modular form 8415.2.a.l
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.