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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 177600.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177600.p1 | 177600di4 | \([0, -1, 0, -16780033, 11815371937]\) | \(127568139540190201/59114336463360\) | \(242132322153922560000000\) | \([2]\) | \(27869184\) | \(3.1816\) | |
| 177600.p2 | 177600di2 | \([0, -1, 0, -8500033, -9535148063]\) | \(16581570075765001/998001000\) | \(4087812096000000000\) | \([2]\) | \(9289728\) | \(2.6323\) | |
| 177600.p3 | 177600di1 | \([0, -1, 0, -500033, -167148063]\) | \(-3375675045001/999000000\) | \(-4091904000000000000\) | \([2]\) | \(4644864\) | \(2.2857\) | \(\Gamma_0(N)\)-optimal |
| 177600.p4 | 177600di3 | \([0, -1, 0, 3699967, 1391051937]\) | \(1367594037332999/995878502400\) | \(-4079118345830400000000\) | \([2]\) | \(13934592\) | \(2.8350\) |
Rank
sage: E.rank()
The elliptic curves in class 177600.p have rank \(1\).
Complex multiplication
The elliptic curves in class 177600.p do not have complex multiplication.Modular form 177600.2.a.p
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.
