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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 1680n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.f3 | 1680n1 | \([0, -1, 0, -40, -80]\) | \(1771561/105\) | \(430080\) | \([2]\) | \(256\) | \(-0.16596\) | \(\Gamma_0(N)\)-optimal |
| 1680.f2 | 1680n2 | \([0, -1, 0, -120, 432]\) | \(47045881/11025\) | \(45158400\) | \([2, 2]\) | \(512\) | \(0.18061\) | |
| 1680.f1 | 1680n3 | \([0, -1, 0, -1800, 30000]\) | \(157551496201/13125\) | \(53760000\) | \([4]\) | \(1024\) | \(0.52719\) | |
| 1680.f4 | 1680n4 | \([0, -1, 0, 280, 2352]\) | \(590589719/972405\) | \(-3982970880\) | \([2]\) | \(1024\) | \(0.52719\) |
Rank
sage: E.rank()
The elliptic curves in class 1680n have rank \(1\).
Complex multiplication
The elliptic curves in class 1680n do not have complex multiplication.Modular form 1680.2.a.n
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.
