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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 16575.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16575.i1 | 16575f5 | \([1, 0, 1, -504351, -137883527]\) | \(908031902324522977/161726530797\) | \(2526977043703125\) | \([2]\) | \(131072\) | \(1.9599\) | |
16575.i2 | 16575f3 | \([1, 0, 1, -34726, -1692277]\) | \(296380748763217/92608836489\) | \(1447013070140625\) | \([2, 2]\) | \(65536\) | \(1.6133\) | |
16575.i3 | 16575f2 | \([1, 0, 1, -13601, 589223]\) | \(17806161424897/668584449\) | \(10446632015625\) | \([2, 2]\) | \(32768\) | \(1.2667\) | |
16575.i4 | 16575f1 | \([1, 0, 1, -13476, 600973]\) | \(17319700013617/25857\) | \(404015625\) | \([2]\) | \(16384\) | \(0.92015\) | \(\Gamma_0(N)\)-optimal |
16575.i5 | 16575f4 | \([1, 0, 1, 5524, 2119223]\) | \(1193377118543/124806800313\) | \(-1950106254890625\) | \([2]\) | \(65536\) | \(1.6133\) | |
16575.i6 | 16575f6 | \([1, 0, 1, 96899, -11432527]\) | \(6439735268725823/7345472585373\) | \(-114773009146453125\) | \([2]\) | \(131072\) | \(1.9599\) |
Rank
sage: E.rank()
The elliptic curves in class 16575.i have rank \(0\).
Complex multiplication
The elliptic curves in class 16575.i do not have complex multiplication.Modular form 16575.2.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.