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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 1638.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1638.p1 | 1638m3 | \([1, -1, 1, -8615, -305585]\) | \(3592121380875/246064\) | \(4843277712\) | \([2]\) | \(1728\) | \(0.91259\) | |
| 1638.p2 | 1638m4 | \([1, -1, 1, -8075, -345977]\) | \(-2958077788875/946054564\) | \(-18621191983212\) | \([2]\) | \(3456\) | \(1.2592\) | |
| 1638.p3 | 1638m1 | \([1, -1, 1, -215, 623]\) | \(40530337875/18264064\) | \(493129728\) | \([6]\) | \(576\) | \(0.36329\) | \(\Gamma_0(N)\)-optimal |
| 1638.p4 | 1638m2 | \([1, -1, 1, 745, 4079]\) | \(1695802078125/1272491584\) | \(-34357272768\) | \([6]\) | \(1152\) | \(0.70986\) |
Rank
sage: E.rank()
The elliptic curves in class 1638.p have rank \(0\).
Complex multiplication
The elliptic curves in class 1638.p do not have complex multiplication.Modular form 1638.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
