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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 5040.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5040.s1 | 5040l4 | \([0, 0, 0, -60483, 5725298]\) | \(32779037733124/315\) | \(235146240\) | \([2]\) | \(8192\) | \(1.1846\) | |
| 5040.s2 | 5040l5 | \([0, 0, 0, -58323, -5403022]\) | \(14695548366242/57421875\) | \(85730400000000\) | \([2]\) | \(16384\) | \(1.5312\) | |
| 5040.s3 | 5040l3 | \([0, 0, 0, -5403, 5402]\) | \(23366901604/13505625\) | \(10081895040000\) | \([2, 2]\) | \(8192\) | \(1.1846\) | |
| 5040.s4 | 5040l2 | \([0, 0, 0, -3783, 89318]\) | \(32082281296/99225\) | \(18517766400\) | \([2, 2]\) | \(4096\) | \(0.83807\) | |
| 5040.s5 | 5040l1 | \([0, 0, 0, -138, 2567]\) | \(-24918016/229635\) | \(-2678462640\) | \([2]\) | \(2048\) | \(0.49150\) | \(\Gamma_0(N)\)-optimal |
| 5040.s6 | 5040l6 | \([0, 0, 0, 21597, 43202]\) | \(746185003198/432360075\) | \(-645510133094400\) | \([2]\) | \(16384\) | \(1.5312\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.s have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.s do not have complex multiplication.Modular form 5040.2.a.s
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
