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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 5850.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5850.d1 | 5850m3 | \([1, -1, 0, -4666392, -3878727984]\) | \(986551739719628473/111045168\) | \(1264873866750000\) | \([2]\) | \(163840\) | \(2.3221\) | |
5850.d2 | 5850m4 | \([1, -1, 0, -526392, 49988016]\) | \(1416134368422073/725251155408\) | \(8261063942069250000\) | \([2]\) | \(163840\) | \(2.3221\) | |
5850.d3 | 5850m2 | \([1, -1, 0, -292392, -60225984]\) | \(242702053576633/2554695936\) | \(29099583396000000\) | \([2, 2]\) | \(81920\) | \(1.9755\) | |
5850.d4 | 5850m1 | \([1, -1, 0, -4392, -2337984]\) | \(-822656953/207028224\) | \(-2358180864000000\) | \([2]\) | \(40960\) | \(1.6290\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5850.d have rank \(0\).
Complex multiplication
The elliptic curves in class 5850.d do not have complex multiplication.Modular form 5850.2.a.d
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.