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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 63525p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63525.bx4 | 63525p1 | \([1, 1, 0, -9551500, -780348125]\) | \(3481467828171481/2005331497785\) | \(55508860524179568515625\) | \([2]\) | \(5529600\) | \(3.0538\) | \(\Gamma_0(N)\)-optimal |
| 63525.bx2 | 63525p2 | \([1, 1, 0, -108786625, -435727901000]\) | \(5143681768032498601/14238434358225\) | \(394128984532678737890625\) | \([2, 2]\) | \(11059200\) | \(3.4004\) | |
| 63525.bx3 | 63525p3 | \([1, 1, 0, -65907250, -782664924125]\) | \(-1143792273008057401/8897444448004035\) | \(-246286962246101191384921875\) | \([2]\) | \(22118400\) | \(3.7470\) | |
| 63525.bx1 | 63525p4 | \([1, 1, 0, -1739428000, -27923449559375]\) | \(21026497979043461623321/161783881875\) | \(4478281493099326171875\) | \([2]\) | \(22118400\) | \(3.7470\) |
Rank
sage: E.rank()
The elliptic curves in class 63525p have rank \(1\).
Complex multiplication
The elliptic curves in class 63525p do not have complex multiplication.Modular form 63525.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
