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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 8085r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8085.n2 | 8085r1 | \([0, 1, 1, 721215, 67404089]\) | \(7196694080651264/4502793796875\) | \(-25957710183018796875\) | \([3]\) | \(181440\) | \(2.4143\) | \(\Gamma_0(N)\)-optimal |
| 8085.n1 | 8085r2 | \([0, 1, 1, -8539785, -10981663486]\) | \(-11947588428895092736/2118439154286675\) | \(-12212380155070978326675\) | \([]\) | \(544320\) | \(2.9636\) |
Rank
sage: E.rank()
The elliptic curves in class 8085r have rank \(1\).
Complex multiplication
The elliptic curves in class 8085r do not have complex multiplication.Modular form 8085.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
