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SageMath
sage: E = EllipticCurve("i1")
sage: E.isogeny_class()
Elliptic curves in class 8619.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 8619.i1 | 8619b5 | [1, 1, 0, -3409409, -2424122748] | [2] | 172032 | |
| 8619.i2 | 8619b3 | [1, 1, 0, -234744, -29790405] | [2, 2] | 86016 | |
| 8619.i3 | 8619b2 | [1, 1, 0, -91939, 10337800] | [2, 2] | 43008 | |
| 8619.i4 | 8619b1 | [1, 1, 0, -91094, 10544487] | [2] | 21504 | \(\Gamma_0(N)\)-optimal |
| 8619.i5 | 8619b4 | [1, 1, 0, 37346, 37254937] | [2] | 86016 | |
| 8619.i6 | 8619b6 | [1, 1, 0, 655041, -200807082] | [4] | 172032 |
Rank
sage: E.rank()
The elliptic curves in class 8619.i have rank \(0\).
Complex multiplication
The elliptic curves in class 8619.i do not have complex multiplication.Modular form 8619.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
