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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 8085u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8085.w3 | 8085u1 | \([1, 0, 1, -9973, -383869]\) | \(932288503609/779625\) | \(91722101625\) | \([2]\) | \(13824\) | \(1.0314\) | \(\Gamma_0(N)\)-optimal |
| 8085.w2 | 8085u2 | \([1, 0, 1, -12178, -202177]\) | \(1697509118089/833765625\) | \(98091692015625\) | \([2, 2]\) | \(27648\) | \(1.3780\) | |
| 8085.w1 | 8085u3 | \([1, 0, 1, -104053, 12770573]\) | \(1058993490188089/13182390375\) | \(1550895045228375\) | \([2]\) | \(55296\) | \(1.7246\) | |
| 8085.w4 | 8085u4 | \([1, 0, 1, 44417, -1537819]\) | \(82375335041831/56396484375\) | \(-6634989990234375\) | \([2]\) | \(55296\) | \(1.7246\) |
Rank
sage: E.rank()
The elliptic curves in class 8085u have rank \(0\).
Complex multiplication
The elliptic curves in class 8085u do not have complex multiplication.Modular form 8085.2.a.u
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.
