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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 162240dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.t4 | 162240dr1 | \([0, -1, 0, 114019, -1026819]\) | \(33165879296/19278675\) | \(-95287789566028800\) | \([2]\) | \(1032192\) | \(1.9474\) | \(\Gamma_0(N)\)-optimal |
| 162240.t3 | 162240dr2 | \([0, -1, 0, -457201, -7767215]\) | \(133649126224/77000625\) | \(6089397203036160000\) | \([2, 2]\) | \(2064384\) | \(2.2940\) | |
| 162240.t2 | 162240dr3 | \([0, -1, 0, -4851201, 4098865185]\) | \(39914580075556/172718325\) | \(54635945366318284800\) | \([2]\) | \(4128768\) | \(2.6406\) | |
| 162240.t1 | 162240dr4 | \([0, -1, 0, -5202721, -4554924479]\) | \(49235161015876/137109375\) | \(43371774950400000000\) | \([2]\) | \(4128768\) | \(2.6406\) |
Rank
sage: E.rank()
The elliptic curves in class 162240dr have rank \(0\).
Complex multiplication
The elliptic curves in class 162240dr do not have complex multiplication.Modular form 162240.2.a.dr
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.
