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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 18150cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18150.cz6 | 18150cp1 | \([1, 0, 0, 771312, -24715008]\) | \(1833318007919/1070530560\) | \(-29632971709440000000\) | \([2]\) | \(552960\) | \(2.4257\) | \(\Gamma_0(N)\)-optimal |
| 18150.cz5 | 18150cp2 | \([1, 0, 0, -3100688, -198955008]\) | \(119102750067601/68309049600\) | \(1890838253412900000000\) | \([2, 2]\) | \(1105920\) | \(2.7723\) | |
| 18150.cz2 | 18150cp3 | \([1, 0, 0, -35770688, -82167985008]\) | \(182864522286982801/463015182960\) | \(12816556883434383750000\) | \([2]\) | \(2211840\) | \(3.1189\) | |
| 18150.cz3 | 18150cp4 | \([1, 0, 0, -32382688, 70634202992]\) | \(135670761487282321/643043610000\) | \(17799859074612656250000\) | \([2, 2]\) | \(2211840\) | \(3.1189\) | |
| 18150.cz1 | 18150cp5 | \([1, 0, 0, -517532188, 4531583855492]\) | \(553808571467029327441/12529687500\) | \(346829776831054687500\) | \([2]\) | \(4423680\) | \(3.4654\) | |
| 18150.cz4 | 18150cp6 | \([1, 0, 0, -15745188, 143123790492]\) | \(-15595206456730321/310672490129100\) | \(-8599613551337476954687500\) | \([2]\) | \(4423680\) | \(3.4654\) |
Rank
sage: E.rank()
The elliptic curves in class 18150cp have rank \(0\).
Complex multiplication
The elliptic curves in class 18150cp do not have complex multiplication.Modular form 18150.2.a.cp
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.
