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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 714.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 714.f1 | 714g5 | \([1, 1, 1, -13718604, -19563199515]\) | \(285531136548675601769470657/17941034271597192\) | \(17941034271597192\) | \([2]\) | \(30720\) | \(2.5791\) | |
| 714.f2 | 714g3 | \([1, 1, 1, -859044, -304722459]\) | \(70108386184777836280897/552468975892674624\) | \(552468975892674624\) | \([2, 2]\) | \(15360\) | \(2.2326\) | |
| 714.f3 | 714g6 | \([1, 1, 1, -292604, -699871003]\) | \(-2770540998624539614657/209924951154647363208\) | \(-209924951154647363208\) | \([2]\) | \(30720\) | \(2.5791\) | |
| 714.f4 | 714g2 | \([1, 1, 1, -90724, 2605541]\) | \(82582985847542515777/44772582831427584\) | \(44772582831427584\) | \([2, 4]\) | \(7680\) | \(1.8860\) | |
| 714.f5 | 714g1 | \([1, 1, 1, -70244, 7127525]\) | \(38331145780597164097/55468445663232\) | \(55468445663232\) | \([8]\) | \(3840\) | \(1.5394\) | \(\Gamma_0(N)\)-optimal |
| 714.f6 | 714g4 | \([1, 1, 1, 349916, 20936165]\) | \(4738217997934888496063/2928751705237796928\) | \(-2928751705237796928\) | \([4]\) | \(15360\) | \(2.2326\) |
Rank
sage: E.rank()
The elliptic curves in class 714.f have rank \(0\).
Complex multiplication
The elliptic curves in class 714.f do not have complex multiplication.Modular form 714.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
