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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 78.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78.a1 | 78a4 | \([1, 1, 0, -20739, 1140957]\) | \(986551739719628473/111045168\) | \(111045168\) | \([4]\) | \(160\) | \(0.96808\) | |
78.a2 | 78a3 | \([1, 1, 0, -2339, -15747]\) | \(1416134368422073/725251155408\) | \(725251155408\) | \([2]\) | \(160\) | \(0.96808\) | |
78.a3 | 78a2 | \([1, 1, 0, -1299, 17325]\) | \(242702053576633/2554695936\) | \(2554695936\) | \([2, 2]\) | \(80\) | \(0.62150\) | |
78.a4 | 78a1 | \([1, 1, 0, -19, 685]\) | \(-822656953/207028224\) | \(-207028224\) | \([2]\) | \(40\) | \(0.27493\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78.a have rank \(0\).
Complex multiplication
The elliptic curves in class 78.a do not have complex multiplication.Modular form 78.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.