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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 14784e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 14784.c4 | 14784e1 | \([0, -1, 0, 111, -17775]\) | \(9148592/8301447\) | \(-136010907648\) | \([2]\) | \(16384\) | \(0.81535\) | \(\Gamma_0(N)\)-optimal |
| 14784.c3 | 14784e2 | \([0, -1, 0, -9569, -348831]\) | \(1478729816932/38900169\) | \(2549361475584\) | \([2, 2]\) | \(32768\) | \(1.1619\) | |
| 14784.c1 | 14784e3 | \([0, -1, 0, -152129, -22787775]\) | \(2970658109581346/2139291\) | \(280401149952\) | \([2]\) | \(65536\) | \(1.5085\) | |
| 14784.c2 | 14784e4 | \([0, -1, 0, -21889, 747649]\) | \(8849350367426/3314597517\) | \(434450925748224\) | \([2]\) | \(65536\) | \(1.5085\) |
Rank
sage: E.rank()
The elliptic curves in class 14784e have rank \(1\).
Complex multiplication
The elliptic curves in class 14784e do not have complex multiplication.Modular form 14784.2.a.e
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.
