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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 19890i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19890.d3 | 19890i1 | \([1, -1, 0, -29070, -1044140]\) | \(3726830856733921/1501644718080\) | \(1094698999480320\) | \([2]\) | \(147456\) | \(1.5837\) | \(\Gamma_0(N)\)-optimal |
| 19890.d2 | 19890i2 | \([1, -1, 0, -213390, 37257556]\) | \(1474074790091785441/32813650022400\) | \(23921150866329600\) | \([2, 2]\) | \(294912\) | \(1.9303\) | |
| 19890.d1 | 19890i3 | \([1, -1, 0, -3395790, 2409418516]\) | \(5940441603429810927841/3044264109120\) | \(2219268535548480\) | \([2]\) | \(589824\) | \(2.2769\) | |
| 19890.d4 | 19890i4 | \([1, -1, 0, 19890, 114193300]\) | \(1193680917131039/7728836230440000\) | \(-5634321611990760000\) | \([2]\) | \(589824\) | \(2.2769\) |
Rank
sage: E.rank()
The elliptic curves in class 19890i have rank \(1\).
Complex multiplication
The elliptic curves in class 19890i do not have complex multiplication.Modular form 19890.2.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.
