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SageMath
E = EllipticCurve("na1")
E.isogeny_class()
Elliptic curves in class 286650na
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.na3 | 286650na1 | \([1, -1, 1, -66380, 42271247]\) | \(-24137569/561600\) | \(-752597711775000000\) | \([2]\) | \(3317760\) | \(2.1109\) | \(\Gamma_0(N)\)-optimal |
| 286650.na2 | 286650na2 | \([1, -1, 1, -2271380, 1312351247]\) | \(967068262369/4928040\) | \(6604044920825625000\) | \([2]\) | \(6635520\) | \(2.4575\) | |
| 286650.na4 | 286650na3 | \([1, -1, 1, 595120, -1116676753]\) | \(17394111071/411937500\) | \(-552035647959960937500\) | \([2]\) | \(9953280\) | \(2.6602\) | |
| 286650.na1 | 286650na4 | \([1, -1, 1, -13186130, -17488801753]\) | \(189208196468929/10860320250\) | \(14553867822816410156250\) | \([2]\) | \(19906560\) | \(3.0068\) |
Rank
sage: E.rank()
The elliptic curves in class 286650na have rank \(1\).
Complex multiplication
The elliptic curves in class 286650na do not have complex multiplication.Modular form 286650.2.a.na
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
