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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 4200d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4200.n5 | 4200d1 | \([0, -1, 0, -383, 12012]\) | \(-24918016/229635\) | \(-57408750000\) | \([2]\) | \(3072\) | \(0.74691\) | \(\Gamma_0(N)\)-optimal |
| 4200.n4 | 4200d2 | \([0, -1, 0, -10508, 417012]\) | \(32082281296/99225\) | \(396900000000\) | \([2, 2]\) | \(6144\) | \(1.0935\) | |
| 4200.n3 | 4200d3 | \([0, -1, 0, -15008, 30012]\) | \(23366901604/13505625\) | \(216090000000000\) | \([2, 2]\) | \(12288\) | \(1.4401\) | |
| 4200.n1 | 4200d4 | \([0, -1, 0, -168008, 26562012]\) | \(32779037733124/315\) | \(5040000000\) | \([2]\) | \(12288\) | \(1.4401\) | |
| 4200.n2 | 4200d5 | \([0, -1, 0, -162008, -24959988]\) | \(14695548366242/57421875\) | \(1837500000000000\) | \([2]\) | \(24576\) | \(1.7866\) | |
| 4200.n6 | 4200d6 | \([0, -1, 0, 59992, 180012]\) | \(746185003198/432360075\) | \(-13835522400000000\) | \([2]\) | \(24576\) | \(1.7866\) |
Rank
sage: E.rank()
The elliptic curves in class 4200d have rank \(0\).
Complex multiplication
The elliptic curves in class 4200d do not have complex multiplication.Modular form 4200.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
