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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 123627r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123627.t6 | 123627r1 | \([1, 0, 1, 40350, 690799]\) | \(103823/63\) | \(-4408763240216727\) | \([2]\) | \(602112\) | \(1.6911\) | \(\Gamma_0(N)\)-optimal |
| 123627.t5 | 123627r2 | \([1, 0, 1, -165695, 5553461]\) | \(7189057/3969\) | \(277752084133653801\) | \([2, 2]\) | \(1204224\) | \(2.0377\) | |
| 123627.t3 | 123627r3 | \([1, 0, 1, -1608010, -780219751]\) | \(6570725617/45927\) | \(3213988402117993983\) | \([2]\) | \(2408448\) | \(2.3842\) | |
| 123627.t2 | 123627r4 | \([1, 0, 1, -2020100, 1103361221]\) | \(13027640977/21609\) | \(1512205791394337361\) | \([2, 2]\) | \(2408448\) | \(2.3842\) | |
| 123627.t4 | 123627r5 | \([1, 0, 1, -1401965, 1791469103]\) | \(-4354703137/17294403\) | \(-1210268701712601334587\) | \([2]\) | \(4816896\) | \(2.7308\) | |
| 123627.t1 | 123627r6 | \([1, 0, 1, -32308715, 70682367599]\) | \(53297461115137/147\) | \(10287114227172363\) | \([2]\) | \(4816896\) | \(2.7308\) |
Rank
sage: E.rank()
The elliptic curves in class 123627r have rank \(1\).
Complex multiplication
The elliptic curves in class 123627r do not have complex multiplication.Modular form 123627.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
