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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 8190k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8190.g5 | 8190k1 | \([1, -1, 0, -3375, -74579]\) | \(5832972054001/4542720\) | \(3311642880\) | \([2]\) | \(8192\) | \(0.75802\) | \(\Gamma_0(N)\)-optimal |
| 8190.g4 | 8190k2 | \([1, -1, 0, -4095, -39875]\) | \(10418796526321/5038160400\) | \(3672818931600\) | \([2, 2]\) | \(16384\) | \(1.1046\) | |
| 8190.g2 | 8190k3 | \([1, -1, 0, -34515, 2448481]\) | \(6237734630203441/82168222500\) | \(59900634202500\) | \([2, 2]\) | \(32768\) | \(1.4512\) | |
| 8190.g6 | 8190k4 | \([1, -1, 0, 14805, -315815]\) | \(492271755328079/342606902820\) | \(-249760432155780\) | \([2]\) | \(32768\) | \(1.4512\) | |
| 8190.g1 | 8190k5 | \([1, -1, 0, -550485, 157342675]\) | \(25306558948218234961/4478906250\) | \(3265122656250\) | \([2]\) | \(65536\) | \(1.7977\) | |
| 8190.g3 | 8190k6 | \([1, -1, 0, -5265, 6444031]\) | \(-22143063655441/24584858584650\) | \(-17922361908209850\) | \([2]\) | \(65536\) | \(1.7977\) |
Rank
sage: E.rank()
The elliptic curves in class 8190k have rank \(1\).
Complex multiplication
The elliptic curves in class 8190k do not have complex multiplication.Modular form 8190.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
