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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 16830cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16830.cu4 | 16830cp1 | \([1, -1, 1, -168557, -26593711]\) | \(726497538898787209/1038579300\) | \(757124309700\) | \([2]\) | \(92160\) | \(1.5507\) | \(\Gamma_0(N)\)-optimal |
| 16830.cu3 | 16830cp2 | \([1, -1, 1, -170087, -26085139]\) | \(746461053445307689/27443694341250\) | \(20006453174771250\) | \([2]\) | \(184320\) | \(1.8973\) | |
| 16830.cu2 | 16830cp3 | \([1, -1, 1, -214592, -10892509]\) | \(1499114720492202169/796539777000000\) | \(580677497433000000\) | \([6]\) | \(276480\) | \(2.1000\) | |
| 16830.cu1 | 16830cp4 | \([1, -1, 1, -1983272, 1067294819]\) | \(1183430669265454849849/10449720703125000\) | \(7617846392578125000\) | \([6]\) | \(552960\) | \(2.4466\) |
Rank
sage: E.rank()
The elliptic curves in class 16830cp have rank \(1\).
Complex multiplication
The elliptic curves in class 16830cp do not have complex multiplication.Modular form 16830.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
