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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 80223n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80223.o4 | 80223n1 | \([1, 0, 1, 74775, 5309599]\) | \(26100282937247/21962862207\) | \(-38908550134295127\) | \([2]\) | \(460800\) | \(1.8710\) | \(\Gamma_0(N)\)-optimal |
| 80223.o3 | 80223n2 | \([1, 0, 1, -366270, 46591411]\) | \(3067396672113073/1245074357241\) | \(2205725173388223201\) | \([2, 2]\) | \(921600\) | \(2.2176\) | |
| 80223.o2 | 80223n3 | \([1, 0, 1, -2702175, -1677306479]\) | \(1231708064988053953/26933399479701\) | \(47714160115658583261\) | \([2]\) | \(1843200\) | \(2.5641\) | |
| 80223.o1 | 80223n4 | \([1, 0, 1, -5087085, 4414289449]\) | \(8218157522273610913/3262914972603\) | \(5780452911779543283\) | \([2]\) | \(1843200\) | \(2.5641\) |
Rank
sage: E.rank()
The elliptic curves in class 80223n have rank \(0\).
Complex multiplication
The elliptic curves in class 80223n do not have complex multiplication.Modular form 80223.2.a.n
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.
