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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 164730.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 164730.d1 | 164730da4 | \([1, 1, 0, -2161003, -1181634947]\) | \(46237740924063961/1806561830400\) | \(43606010834046297600\) | \([2]\) | \(4478976\) | \(2.5357\) | |
| 164730.d2 | 164730da2 | \([1, 1, 0, -318628, 68600728]\) | \(148212258825961/1218375000\) | \(29408610630375000\) | \([2]\) | \(1492992\) | \(1.9864\) | |
| 164730.d3 | 164730da1 | \([1, 1, 0, -6508, 2493712]\) | \(-1263214441/110808000\) | \(-2674635745752000\) | \([2]\) | \(746496\) | \(1.6398\) | \(\Gamma_0(N)\)-optimal |
| 164730.d4 | 164730da3 | \([1, 1, 0, 58517, -66992003]\) | \(918046641959/80912056320\) | \(-1953020342355886080\) | \([2]\) | \(2239488\) | \(2.1891\) |
Rank
sage: E.rank()
The elliptic curves in class 164730.d have rank \(0\).
Complex multiplication
The elliptic curves in class 164730.d do not have complex multiplication.Modular form 164730.2.a.d
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
