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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 47040.dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.dp1 | 47040fi4 | \([0, -1, 0, -219585, 39678465]\) | \(303735479048/105\) | \(404787855360\) | \([4]\) | \(294912\) | \(1.5831\) | |
| 47040.dp2 | 47040fi2 | \([0, -1, 0, -13785, 617625]\) | \(601211584/11025\) | \(5312840601600\) | \([2, 2]\) | \(147456\) | \(1.2365\) | |
| 47040.dp3 | 47040fi1 | \([0, -1, 0, -1780, -13838]\) | \(82881856/36015\) | \(271176239040\) | \([2]\) | \(73728\) | \(0.88997\) | \(\Gamma_0(N)\)-optimal |
| 47040.dp4 | 47040fi3 | \([0, -1, 0, -65, 1778337]\) | \(-8/354375\) | \(-1366159011840000\) | \([2]\) | \(294912\) | \(1.5831\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.dp have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.dp do not have complex multiplication.Modular form 47040.2.a.dp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
