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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 193200.gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.gp1 | 193200bh3 | \([0, 1, 0, -1155008, 156743988]\) | \(2662558086295801/1374177967680\) | \(87947389931520000000\) | \([2]\) | \(4976640\) | \(2.5190\) | |
| 193200.gp2 | 193200bh1 | \([0, 1, 0, -645008, -199596012]\) | \(463702796512201/15214500\) | \(973728000000000\) | \([2]\) | \(1658880\) | \(1.9697\) | \(\Gamma_0(N)\)-optimal |
| 193200.gp3 | 193200bh2 | \([0, 1, 0, -617008, -217684012]\) | \(-405897921250921/84358968750\) | \(-5398974000000000000\) | \([2]\) | \(3317760\) | \(2.3163\) | |
| 193200.gp4 | 193200bh4 | \([0, 1, 0, 4332992, 1221415988]\) | \(140574743422291079/91397357868600\) | \(-5849430903590400000000\) | \([2]\) | \(9953280\) | \(2.8656\) |
Rank
sage: E.rank()
The elliptic curves in class 193200.gp have rank \(0\).
Complex multiplication
The elliptic curves in class 193200.gp do not have complex multiplication.Modular form 193200.2.a.gp
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
