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SageMath
E = EllipticCurve("dj1")
E.isogeny_class()
Elliptic curves in class 339150dj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 339150.dj3 | 339150dj1 | \([1, 1, 1, -8588, -229219]\) | \(4483146738169/1186753680\) | \(18543026250000\) | \([2]\) | \(884736\) | \(1.2545\) | \(\Gamma_0(N)\)-optimal |
| 339150.dj2 | 339150dj2 | \([1, 1, 1, -49088, 3982781]\) | \(837201991720249/41408180100\) | \(647002814062500\) | \([2, 2]\) | \(1769472\) | \(1.6011\) | |
| 339150.dj1 | 339150dj3 | \([1, 1, 1, -775838, 262705781]\) | \(3305345506018293529/8724633750\) | \(136322402343750\) | \([2]\) | \(3538944\) | \(1.9476\) | |
| 339150.dj4 | 339150dj4 | \([1, 1, 1, 29662, 15637781]\) | \(184715807453351/6857260351830\) | \(-107144692997343750\) | \([2]\) | \(3538944\) | \(1.9476\) |
Rank
sage: E.rank()
The elliptic curves in class 339150dj have rank \(1\).
Complex multiplication
The elliptic curves in class 339150dj do not have complex multiplication.Modular form 339150.2.a.dj
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.
