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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 9744.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9744.d1 | 9744h4 | \([0, -1, 0, -92688, 6982848]\) | \(21500025903924625/7344878367708\) | \(30084621794131968\) | \([2]\) | \(69120\) | \(1.8639\) | |
9744.d2 | 9744h2 | \([0, -1, 0, -37968, -2834496]\) | \(1477843225692625/274663872\) | \(1125023219712\) | \([2]\) | \(23040\) | \(1.3146\) | |
9744.d3 | 9744h1 | \([0, -1, 0, -2128, -53312]\) | \(-260305116625/157151232\) | \(-643691446272\) | \([2]\) | \(11520\) | \(0.96804\) | \(\Gamma_0(N)\)-optimal |
9744.d4 | 9744h3 | \([0, -1, 0, 17072, 748480]\) | \(134335727363375/137728390128\) | \(-564135485964288\) | \([2]\) | \(34560\) | \(1.5173\) |
Rank
sage: E.rank()
The elliptic curves in class 9744.d have rank \(0\).
Complex multiplication
The elliptic curves in class 9744.d do not have complex multiplication.Modular form 9744.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.