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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 32448cg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32448.bm4 | 32448cg1 | \([0, -1, 0, -211137, -779529375]\) | \(-822656953/207028224\) | \(-261956749192650031104\) | \([2]\) | \(1290240\) | \(2.5971\) | \(\Gamma_0(N)\)-optimal |
| 32448.bm3 | 32448cg2 | \([0, -1, 0, -14055617, -20092578975]\) | \(242702053576633/2554695936\) | \(3232505354295240032256\) | \([2, 2]\) | \(2580480\) | \(2.9437\) | |
| 32448.bm2 | 32448cg3 | \([0, -1, 0, -25304257, 16625231713]\) | \(1416134368422073/725251155408\) | \(917674080123940522426368\) | \([2]\) | \(5160960\) | \(3.2903\) | |
| 32448.bm1 | 32448cg4 | \([0, -1, 0, -224318657, -1293067075743]\) | \(986551739719628473/111045168\) | \(140507562982483427328\) | \([2]\) | \(5160960\) | \(3.2903\) |
Rank
sage: E.rank()
The elliptic curves in class 32448cg have rank \(0\).
Complex multiplication
The elliptic curves in class 32448cg do not have complex multiplication.Modular form 32448.2.a.cg
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 Cremona 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 Cremona labels.
