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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 975.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 975.f1 | 975g3 | \([1, 0, 0, -1738, -28033]\) | \(37159393753/1053\) | \(16453125\) | \([2]\) | \(512\) | \(0.48635\) | |
| 975.f2 | 975g4 | \([1, 0, 0, -488, 3717]\) | \(822656953/85683\) | \(1338796875\) | \([2]\) | \(512\) | \(0.48635\) | |
| 975.f3 | 975g2 | \([1, 0, 0, -113, -408]\) | \(10218313/1521\) | \(23765625\) | \([2, 2]\) | \(256\) | \(0.13978\) | |
| 975.f4 | 975g1 | \([1, 0, 0, 12, -33]\) | \(12167/39\) | \(-609375\) | \([2]\) | \(128\) | \(-0.20679\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 975.f have rank \(0\).
Complex multiplication
The elliptic curves in class 975.f do not have complex multiplication.Modular form 975.2.a.f
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.
