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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1638i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1638.d3 | 1638i1 | \([1, -1, 0, -63, 189]\) | \(38272753/4368\) | \(3184272\) | \([2]\) | \(384\) | \(-0.019868\) | \(\Gamma_0(N)\)-optimal |
| 1638.d2 | 1638i2 | \([1, -1, 0, -243, -1215]\) | \(2181825073/298116\) | \(217326564\) | \([2, 2]\) | \(768\) | \(0.32670\) | |
| 1638.d1 | 1638i3 | \([1, -1, 0, -3753, -87561]\) | \(8020417344913/187278\) | \(136525662\) | \([2]\) | \(1536\) | \(0.67328\) | |
| 1638.d4 | 1638i4 | \([1, -1, 0, 387, -6885]\) | \(8780064047/32388174\) | \(-23610978846\) | \([2]\) | \(1536\) | \(0.67328\) |
Rank
sage: E.rank()
The elliptic curves in class 1638i have rank \(1\).
Complex multiplication
The elliptic curves in class 1638i do not have complex multiplication.Modular form 1638.2.a.i
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.
