Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 99372.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 99372.p1 | 99372g4 | \([0, -1, 0, -15140428, 22680444760]\) | \(2640279346000/3087\) | \(448771169548945152\) | \([2]\) | \(3981312\) | \(2.6702\) | |
| 99372.p2 | 99372g3 | \([0, -1, 0, -938513, 360715146]\) | \(-10061824000/352947\) | \(-3206843982401837232\) | \([2]\) | \(1990656\) | \(2.3237\) | |
| 99372.p3 | 99372g2 | \([0, -1, 0, -234628, 14089048]\) | \(9826000/5103\) | \(741846219050297088\) | \([2]\) | \(1327104\) | \(2.1209\) | |
| 99372.p4 | 99372g1 | \([0, -1, 0, 55207, 1684110]\) | \(2048000/1323\) | \(-12020656327203888\) | \([2]\) | \(663552\) | \(1.7744\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 99372.p have rank \(1\).
Complex multiplication
The elliptic curves in class 99372.p do not have complex multiplication.Modular form 99372.2.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.
