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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 67280f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67280.r3 | 67280f1 | \([0, 0, 0, -203522, -35339661]\) | \(97960237056/725\) | \(6899950523600\) | \([2]\) | \(215040\) | \(1.6399\) | \(\Gamma_0(N)\)-optimal |
| 67280.r2 | 67280f2 | \([0, 0, 0, -207727, -33803154]\) | \(6509904336/525625\) | \(80039426073760000\) | \([2, 2]\) | \(430080\) | \(1.9865\) | |
| 67280.r4 | 67280f3 | \([0, 0, 0, 212773, -153309254]\) | \(1748981916/17682025\) | \(-10770105172485145600\) | \([2]\) | \(860160\) | \(2.3330\) | |
| 67280.r1 | 67280f4 | \([0, 0, 0, -695507, 184039394]\) | \(61085802564/11328125\) | \(6899950523600000000\) | \([4]\) | \(860160\) | \(2.3330\) |
Rank
sage: E.rank()
The elliptic curves in class 67280f have rank \(0\).
Complex multiplication
The elliptic curves in class 67280f do not have complex multiplication.Modular form 67280.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
