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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 123786bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123786.bm4 | 123786bn1 | \([1, -1, 1, -92939, 227700123]\) | \(-822656953/207028224\) | \(-22342105640001798144\) | \([2]\) | \(4055040\) | \(2.3920\) | \(\Gamma_0(N)\)-optimal |
123786.bm3 | 123786bn2 | \([1, -1, 1, -6187019, 5870818203]\) | \(242702053576633/2554695936\) | \(275698092643615938816\) | \([2, 2]\) | \(8110080\) | \(2.7386\) | |
123786.bm2 | 123786bn3 | \([1, -1, 1, -11138459, -4850039685]\) | \(1416134368422073/725251155408\) | \(78267772464004219092048\) | \([2]\) | \(16220160\) | \(3.0851\) | |
123786.bm1 | 123786bn4 | \([1, -1, 1, -98740859, 377678104251]\) | \(986551739719628473/111045168\) | \(11983790549581042608\) | \([2]\) | \(16220160\) | \(3.0851\) |
Rank
sage: E.rank()
The elliptic curves in class 123786bn have rank \(1\).
Complex multiplication
The elliptic curves in class 123786bn do not have complex multiplication.Modular form 123786.2.a.bn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.