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SageMath
E = EllipticCurve("hm1")
E.isogeny_class()
Elliptic curves in class 270480hm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 270480.hm3 | 270480hm1 | \([0, 1, 0, -4602783460, -120194466204100]\) | \(358061097267989271289240144/176126855625\) | \(5304613999980960000\) | \([2]\) | \(106168320\) | \(3.8301\) | \(\Gamma_0(N)\)-optimal |
| 270480.hm2 | 270480hm2 | \([0, 1, 0, -4602807960, -120193122692700]\) | \(89516703758060574923008036/1985322833430374025\) | \(239176955934976075435238400\) | \([2, 2]\) | \(212336640\) | \(4.1767\) | |
| 270480.hm1 | 270480hm3 | \([0, 1, 0, -4767516560, -111128681367180]\) | \(49737293673675178002921218/6641736806881023047235\) | \(1600294284477942744031540254720\) | \([4]\) | \(424673280\) | \(4.5233\) | |
| 270480.hm4 | 270480hm4 | \([0, 1, 0, -4438491360, -129171578896620]\) | \(-40133926989810174413190818/6689384645060302103835\) | \(-1611774800090520539574443857920\) | \([2]\) | \(424673280\) | \(4.5233\) |
Rank
sage: E.rank()
The elliptic curves in class 270480hm have rank \(1\).
Complex multiplication
The elliptic curves in class 270480hm do not have complex multiplication.Modular form 270480.2.a.hm
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.
