Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 8085.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8085.j1 | 8085c2 | \([0, -1, 1, -174281, 32066306]\) | \(-11947588428895092736/2118439154286675\) | \(-103803518560047075\) | \([]\) | \(77760\) | \(1.9907\) | |
| 8085.j2 | 8085c1 | \([0, -1, 1, 14719, -200719]\) | \(7196694080651264/4502793796875\) | \(-220636896046875\) | \([]\) | \(25920\) | \(1.4414\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8085.j have rank \(0\).
Complex multiplication
The elliptic curves in class 8085.j do not have complex multiplication.Modular form 8085.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
