Show commands:
SageMath
E = EllipticCurve("pd1")
E.isogeny_class()
Elliptic curves in class 158400pd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 158400.nv3 | 158400pd1 | \([0, 0, 0, -2508300, -323882000]\) | \(15781142246787/8722841600\) | \(964676498227200000000\) | \([2]\) | \(7962624\) | \(2.7170\) | \(\Gamma_0(N)\)-optimal |
| 158400.nv4 | 158400pd2 | \([0, 0, 0, 9779700, -2560298000]\) | \(935355271080573/566899520000\) | \(-62694551715840000000000\) | \([2]\) | \(15925248\) | \(3.0636\) | |
| 158400.nv1 | 158400pd3 | \([0, 0, 0, -154572300, -739681578000]\) | \(5066026756449723/11000000\) | \(886837248000000000000\) | \([2]\) | \(23887872\) | \(3.2663\) | |
| 158400.nv2 | 158400pd4 | \([0, 0, 0, -152844300, -757027242000]\) | \(-4898016158612283/236328125000\) | \(-19053144000000000000000000\) | \([2]\) | \(47775744\) | \(3.6129\) |
Rank
sage: E.rank()
The elliptic curves in class 158400pd have rank \(1\).
Complex multiplication
The elliptic curves in class 158400pd do not have complex multiplication.Modular form 158400.2.a.pd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
