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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 34650cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.bp4 | 34650cb1 | \([1, -1, 0, -1057077, 418583781]\) | \(1433528304665250149/162339408\) | \(14793178554000\) | \([2]\) | \(307200\) | \(1.9510\) | \(\Gamma_0(N)\)-optimal |
| 34650.bp3 | 34650cb2 | \([1, -1, 0, -1059777, 416340081]\) | \(1444540994277943589/15251205665388\) | \(1389766116258481500\) | \([2]\) | \(614400\) | \(2.2976\) | |
| 34650.bp2 | 34650cb3 | \([1, -1, 0, -3905802, -2531346444]\) | \(72313087342699809269/11447096545640448\) | \(1043116672721485824000\) | \([2]\) | \(1536000\) | \(2.7557\) | |
| 34650.bp1 | 34650cb4 | \([1, -1, 0, -59893002, -178387141644]\) | \(260744057755293612689909/8504954620259328\) | \(775013989771131264000\) | \([2]\) | \(3072000\) | \(3.1023\) |
Rank
sage: E.rank()
The elliptic curves in class 34650cb have rank \(0\).
Complex multiplication
The elliptic curves in class 34650cb do not have complex multiplication.Modular form 34650.2.a.cb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
