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SageMath
E = EllipticCurve("gs1")
E.isogeny_class()
Elliptic curves in class 81600gs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81600.j4 | 81600gs1 | \([0, -1, 0, -2023633, -33634134863]\) | \(-3579968623693264/1906997690433375\) | \(-488191408750944000000000\) | \([2]\) | \(10321920\) | \(3.2245\) | \(\Gamma_0(N)\)-optimal |
| 81600.j3 | 81600gs2 | \([0, -1, 0, -169065633, -837273196863]\) | \(521902963282042184836/6241849278890625\) | \(6391653661584000000000000\) | \([2, 2]\) | \(20643840\) | \(3.5711\) | |
| 81600.j2 | 81600gs3 | \([0, -1, 0, -313565633, 806125303137]\) | \(1664865424893526702418/826424127435466125\) | \(1692516612987834624000000000\) | \([2]\) | \(41287680\) | \(3.9177\) | |
| 81600.j1 | 81600gs4 | \([0, -1, 0, -2697237633, -53916244336863]\) | \(1059623036730633329075378/154307373046875\) | \(316021500000000000000000\) | \([2]\) | \(41287680\) | \(3.9177\) |
Rank
sage: E.rank()
The elliptic curves in class 81600gs have rank \(1\).
Complex multiplication
The elliptic curves in class 81600gs do not have complex multiplication.Modular form 81600.2.a.gs
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.
