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SageMath
E = EllipticCurve("gf1")
E.isogeny_class()
Elliptic curves in class 286110gf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.gf3 | 286110gf1 | \([1, -1, 1, -136233932, 611957778239]\) | \(15891267085572193561/3334993530000\) | \(58683505968352927530000\) | \([2]\) | \(42467328\) | \(3.3649\) | \(\Gamma_0(N)\)-optimal |
| 286110.gf2 | 286110gf2 | \([1, -1, 1, -151267712, 468571598111]\) | \(21754112339458491481/7199734626562500\) | \(126688602579819009314062500\) | \([2, 2]\) | \(84934656\) | \(3.7115\) | |
| 286110.gf4 | 286110gf3 | \([1, -1, 1, 437260558, 3224296369559]\) | \(525440531549759128199/559322204589843750\) | \(-9841994485452693786621093750\) | \([2]\) | \(169869312\) | \(4.0580\) | |
| 286110.gf1 | 286110gf4 | \([1, -1, 1, -980336462, -11464049106889]\) | \(5921450764096952391481/200074809015963750\) | \(3520573921175007748383213750\) | \([2]\) | \(169869312\) | \(4.0580\) |
Rank
sage: E.rank()
The elliptic curves in class 286110gf have rank \(1\).
Complex multiplication
The elliptic curves in class 286110gf do not have complex multiplication.Modular form 286110.2.a.gf
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.
