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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 61200.do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61200.do1 | 61200fh6 | \([0, 0, 0, -99878475, -384198767750]\) | \(2361739090258884097/5202\) | \(242704512000000\) | \([2]\) | \(3145728\) | \(2.8942\) | |
| 61200.do2 | 61200fh4 | \([0, 0, 0, -6242475, -6002963750]\) | \(576615941610337/27060804\) | \(1262548871424000000\) | \([2, 2]\) | \(1572864\) | \(2.5477\) | |
| 61200.do3 | 61200fh5 | \([0, 0, 0, -5918475, -6653879750]\) | \(-491411892194497/125563633938\) | \(-5858296905011328000000\) | \([2]\) | \(3145728\) | \(2.8942\) | |
| 61200.do4 | 61200fh2 | \([0, 0, 0, -410475, -83483750]\) | \(163936758817/30338064\) | \(1415452713984000000\) | \([2, 2]\) | \(786432\) | \(2.2011\) | |
| 61200.do5 | 61200fh1 | \([0, 0, 0, -122475, 15300250]\) | \(4354703137/352512\) | \(16446799872000000\) | \([2]\) | \(393216\) | \(1.8545\) | \(\Gamma_0(N)\)-optimal |
| 61200.do6 | 61200fh3 | \([0, 0, 0, 813525, -486179750]\) | \(1276229915423/2927177028\) | \(-136570371418368000000\) | \([2]\) | \(1572864\) | \(2.5477\) |
Rank
sage: E.rank()
The elliptic curves in class 61200.do have rank \(0\).
Complex multiplication
The elliptic curves in class 61200.do do not have complex multiplication.Modular form 61200.2.a.do
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
