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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 46800cx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 46800.de6 | 46800cx1 | \([0, 0, 0, 53925, 1930250]\) | \(371694959/249600\) | \(-11645337600000000\) | \([2]\) | \(294912\) | \(1.7715\) | \(\Gamma_0(N)\)-optimal |
| 46800.de5 | 46800cx2 | \([0, 0, 0, -234075, 16042250]\) | \(30400540561/15210000\) | \(709637760000000000\) | \([2, 2]\) | \(589824\) | \(2.1181\) | |
| 46800.de3 | 46800cx3 | \([0, 0, 0, -2034075, -1105357750]\) | \(19948814692561/231344100\) | \(10793590329600000000\) | \([2, 2]\) | \(1179648\) | \(2.4647\) | |
| 46800.de2 | 46800cx4 | \([0, 0, 0, -3042075, 2040610250]\) | \(66730743078481/60937500\) | \(2843100000000000000\) | \([2]\) | \(1179648\) | \(2.4647\) | |
| 46800.de4 | 46800cx5 | \([0, 0, 0, -414075, -2817697750]\) | \(-168288035761/73415764890\) | \(-3425285926707840000000\) | \([2]\) | \(2359296\) | \(2.8112\) | |
| 46800.de1 | 46800cx6 | \([0, 0, 0, -32454075, -71162617750]\) | \(81025909800741361/11088090\) | \(517325927040000000\) | \([2]\) | \(2359296\) | \(2.8112\) |
Rank
sage: E.rank()
The elliptic curves in class 46800cx have rank \(1\).
Complex multiplication
The elliptic curves in class 46800cx do not have complex multiplication.Modular form 46800.2.a.cx
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
