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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 4680f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4680.g5 | 4680f1 | \([0, 0, 0, -83658, -10080007]\) | \(-5551350318708736/550618236675\) | \(-6422411112577200\) | \([4]\) | \(30720\) | \(1.7737\) | \(\Gamma_0(N)\)-optimal |
| 4680.g4 | 4680f2 | \([0, 0, 0, -1368903, -616458598]\) | \(1520107298839022416/13013105625\) | \(2428557824160000\) | \([2, 2]\) | \(61440\) | \(2.1203\) | |
| 4680.g1 | 4680f3 | \([0, 0, 0, -21902403, -39453520498]\) | \(1556580279686303289604/114075\) | \(85156531200\) | \([2]\) | \(122880\) | \(2.4669\) | |
| 4680.g3 | 4680f4 | \([0, 0, 0, -1399323, -587626522]\) | \(405929061432816484/35083409765625\) | \(26189625056400000000\) | \([2, 2]\) | \(122880\) | \(2.4669\) | |
| 4680.g2 | 4680f5 | \([0, 0, 0, -4811043, 3391121342]\) | \(8248670337458940482/1446075439453125\) | \(2158979062500000000000\) | \([2]\) | \(245760\) | \(2.8134\) | |
| 4680.g6 | 4680f6 | \([0, 0, 0, 1525677, -2721121522]\) | \(263059523447441758/2294739983908125\) | \(-3426028438054959360000\) | \([2]\) | \(245760\) | \(2.8134\) |
Rank
sage: E.rank()
The elliptic curves in class 4680f have rank \(1\).
Complex multiplication
The elliptic curves in class 4680f do not have complex multiplication.Modular form 4680.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
