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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 187200en
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.hy3 | 187200en1 | \([0, 0, 0, -194700, 32074000]\) | \(273359449/9360\) | \(27948810240000000\) | \([2]\) | \(1179648\) | \(1.9276\) | \(\Gamma_0(N)\)-optimal |
| 187200.hy2 | 187200en2 | \([0, 0, 0, -482700, -84854000]\) | \(4165509529/1368900\) | \(4087513497600000000\) | \([2, 2]\) | \(2359296\) | \(2.2742\) | |
| 187200.hy4 | 187200en3 | \([0, 0, 0, 1389300, -582806000]\) | \(99317171591/106616250\) | \(-318354416640000000000\) | \([2]\) | \(4718592\) | \(2.6208\) | |
| 187200.hy1 | 187200en4 | \([0, 0, 0, -6962700, -7070294000]\) | \(12501706118329/2570490\) | \(7675442012160000000\) | \([2]\) | \(4718592\) | \(2.6208\) |
Rank
sage: E.rank()
The elliptic curves in class 187200en have rank \(1\).
Complex multiplication
The elliptic curves in class 187200en do not have complex multiplication.Modular form 187200.2.a.en
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.
