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SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 374850.ir
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 374850.ir1 | 374850ir6 | \([1, -1, 1, -151247609330, -22640193943535703]\) | \(285531136548675601769470657/17941034271597192\) | \(24042701815671119255191125000\) | \([2]\) | \(1509949440\) | \(4.9061\) | |
| 374850.ir2 | 374850ir4 | \([1, -1, 1, -9470960330, -352337614139703]\) | \(70108386184777836280897/552468975892674624\) | \(740361266174331481493961000000\) | \([2, 2]\) | \(754974720\) | \(4.5595\) | |
| 374850.ir3 | 374850ir5 | \([1, -1, 1, -3225959330, -810046227431703]\) | \(-2770540998624539614657/209924951154647363208\) | \(-281319511900758991644191815125000\) | \([2]\) | \(1509949440\) | \(4.9061\) | |
| 374850.ir4 | 374850ir2 | \([1, -1, 1, -1000232330, 3060249828297]\) | \(82582985847542515777/44772582831427584\) | \(59999543071917824563776000000\) | \([2, 2]\) | \(377487360\) | \(4.2130\) | |
| 374850.ir5 | 374850ir1 | \([1, -1, 1, -774440330, 8285076708297]\) | \(38331145780597164097/55468445663232\) | \(74333022225541890048000000\) | \([2]\) | \(188743680\) | \(3.8664\) | \(\Gamma_0(N)\)-optimal |
| 374850.ir6 | 374850ir3 | \([1, -1, 1, 3857823670, 24066483972297]\) | \(4738217997934888496063/2928751705237796928\) | \(-3924807392662206642191817000000\) | \([2]\) | \(754974720\) | \(4.5595\) |
Rank
sage: E.rank()
The elliptic curves in class 374850.ir have rank \(0\).
Complex multiplication
The elliptic curves in class 374850.ir do not have complex multiplication.Modular form 374850.2.a.ir
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.
