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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 6270.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6270.f1 | 6270b4 | \([1, 1, 0, -1765632, 902286864]\) | \(608729950623321661295881/206910000\) | \(206910000\) | \([4]\) | \(81920\) | \(1.8584\) | |
| 6270.f2 | 6270b3 | \([1, 1, 0, -113952, 13092336]\) | \(163642280484049092361/20113533886176720\) | \(20113533886176720\) | \([2]\) | \(81920\) | \(1.8584\) | |
| 6270.f3 | 6270b2 | \([1, 1, 0, -110352, 14063616]\) | \(148617683376642237961/2739951878400\) | \(2739951878400\) | \([2, 2]\) | \(40960\) | \(1.5119\) | |
| 6270.f4 | 6270b1 | \([1, 1, 0, -6672, 232704]\) | \(-32854399024748041/4942639595520\) | \(-4942639595520\) | \([2]\) | \(20480\) | \(1.1653\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6270.f have rank \(0\).
Complex multiplication
The elliptic curves in class 6270.f do not have complex multiplication.Modular form 6270.2.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.
