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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 28314r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28314.u2 | 28314r1 | \([1, -1, 0, -7101, -329103]\) | \(-30664297/18876\) | \(-24377749382844\) | \([2]\) | \(76800\) | \(1.2706\) | \(\Gamma_0(N)\)-optimal |
28314.u1 | 28314r2 | \([1, -1, 0, -126891, -17363241]\) | \(174958262857/33462\) | \(43215101178678\) | \([2]\) | \(153600\) | \(1.6172\) |
Rank
sage: E.rank()
The elliptic curves in class 28314r have rank \(1\).
Complex multiplication
The elliptic curves in class 28314r do not have complex multiplication.Modular form 28314.2.a.r
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.