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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 178752.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
178752.r1 | 178752de3 | \([0, -1, 0, -56923169, 165322254273]\) | \(661397832743623417/443352042\) | \(13673411219097649152\) | \([4]\) | \(11796480\) | \(2.9880\) | |
178752.r2 | 178752de2 | \([0, -1, 0, -3579809, 2550325569]\) | \(164503536215257/4178071044\) | \(128855803233712472064\) | \([2, 2]\) | \(5898240\) | \(2.6414\) | |
178752.r3 | 178752de1 | \([0, -1, 0, -506529, -81016767]\) | \(466025146777/177366672\) | \(5470161887331090432\) | \([2]\) | \(2949120\) | \(2.2948\) | \(\Gamma_0(N)\)-optimal |
178752.r4 | 178752de4 | \([0, -1, 0, 591071, 8135133889]\) | \(740480746823/927484650666\) | \(-28604535056945442717696\) | \([2]\) | \(11796480\) | \(2.9880\) |
Rank
sage: E.rank()
The elliptic curves in class 178752.r have rank \(0\).
Complex multiplication
The elliptic curves in class 178752.r do not have complex multiplication.Modular form 178752.2.a.r
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.