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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 122010p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122010.t1 | 122010p1 | \([1, 0, 1, -181669, -8084464]\) | \(16431620361967/8811970560\) | \(355594796873809920\) | \([2]\) | \(1935360\) | \(2.0589\) | \(\Gamma_0(N)\)-optimal |
122010.t2 | 122010p2 | \([1, 0, 1, 696411, -63227888]\) | \(925633609502993/578543731200\) | \(-23346326361158438400\) | \([2]\) | \(3870720\) | \(2.4055\) |
Rank
sage: E.rank()
The elliptic curves in class 122010p have rank \(1\).
Complex multiplication
The elliptic curves in class 122010p do not have complex multiplication.Modular form 122010.2.a.p
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.