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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 154560gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 154560.du3 | 154560gk1 | \([0, -1, 0, -8135905, 9569257825]\) | \(-227196402372228188089/19338934824115200\) | \(-5069585730532854988800\) | \([2]\) | \(8847360\) | \(2.9096\) | \(\Gamma_0(N)\)-optimal |
| 154560.du2 | 154560gk2 | \([0, -1, 0, -132725985, 588589195617]\) | \(986396822567235411402169/6336721794060000\) | \(1661133597982064640000\) | \([2]\) | \(17694720\) | \(3.2562\) | |
| 154560.du4 | 154560gk3 | \([0, -1, 0, 48234335, 337640737]\) | \(47342661265381757089751/27397579603968000000\) | \(-7182111107702587392000000\) | \([2]\) | \(26542080\) | \(3.4589\) | |
| 154560.du1 | 154560gk4 | \([0, -1, 0, -192938145, 2894069025]\) | \(3029968325354577848895529/1753440696000000000000\) | \(459653957812224000000000000\) | \([2]\) | \(53084160\) | \(3.8055\) |
Rank
sage: E.rank()
The elliptic curves in class 154560gk have rank \(0\).
Complex multiplication
The elliptic curves in class 154560gk do not have complex multiplication.Modular form 154560.2.a.gk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
