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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 65598bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
65598.bm4 | 65598bl1 | \([1, 0, 0, -16417, 16901705]\) | \(-822656953/207028224\) | \(-123145215740411904\) | \([2]\) | \(967680\) | \(1.9586\) | \(\Gamma_0(N)\)-optimal |
65598.bm3 | 65598bl2 | \([1, 0, 0, -1092897, 435652425]\) | \(242702053576633/2554695936\) | \(1519592720796723456\) | \([2, 2]\) | \(1935360\) | \(2.3052\) | |
65598.bm2 | 65598bl3 | \([1, 0, 0, -1967537, -360444903]\) | \(1416134368422073/725251155408\) | \(431396300818873669968\) | \([2]\) | \(3870720\) | \(2.6517\) | |
65598.bm1 | 65598bl4 | \([1, 0, 0, -17441937, 28036101753]\) | \(986551739719628473/111045168\) | \(66052255610762928\) | \([2]\) | \(3870720\) | \(2.6517\) |
Rank
sage: E.rank()
The elliptic curves in class 65598bl have rank \(0\).
Complex multiplication
The elliptic curves in class 65598bl do not have complex multiplication.Modular form 65598.2.a.bl
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.