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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 82a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82.a2 | 82a1 | \([1, 0, 1, -2, 0]\) | \(389017/164\) | \(164\) | \([2]\) | \(4\) | \(-0.87815\) | \(\Gamma_0(N)\)-optimal |
82.a1 | 82a2 | \([1, 0, 1, -12, -16]\) | \(169112377/3362\) | \(3362\) | \([2]\) | \(8\) | \(-0.53158\) |
Rank
sage: E.rank()
The elliptic curves in class 82a have rank \(1\).
Complex multiplication
The elliptic curves in class 82a do not have complex multiplication.Modular form 82.2.a.a
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.