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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 8619.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8619.i1 | 8619b5 | \([1, 1, 0, -3409409, -2424122748]\) | \(908031902324522977/161726530797\) | \(780623074389736773\) | \([2]\) | \(172032\) | \(2.4376\) | |
8619.i2 | 8619b3 | \([1, 1, 0, -234744, -29790405]\) | \(296380748763217/92608836489\) | \(447005165444633601\) | \([2, 2]\) | \(86016\) | \(2.0911\) | |
8619.i3 | 8619b2 | \([1, 1, 0, -91939, 10337800]\) | \(17806161424897/668584449\) | \(3227129435693241\) | \([2, 2]\) | \(43008\) | \(1.7445\) | |
8619.i4 | 8619b1 | \([1, 1, 0, -91094, 10544487]\) | \(17319700013617/25857\) | \(124806800313\) | \([2]\) | \(21504\) | \(1.3979\) | \(\Gamma_0(N)\)-optimal |
8619.i5 | 8619b4 | \([1, 1, 0, 37346, 37254937]\) | \(1193377118543/124806800313\) | \(-602418587011991217\) | \([2]\) | \(86016\) | \(2.0911\) | |
8619.i6 | 8619b6 | \([1, 1, 0, 655041, -200807082]\) | \(6439735268725823/7345472585373\) | \(-35455193184331664757\) | \([4]\) | \(172032\) | \(2.4376\) |
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.