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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 34650cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.dg4 | 34650cz1 | \([1, -1, 1, 2245, -335253]\) | \(109902239/4312000\) | \(-49116375000000\) | \([2]\) | \(110592\) | \(1.3070\) | \(\Gamma_0(N)\)-optimal |
| 34650.dg2 | 34650cz2 | \([1, -1, 1, -60755, -5501253]\) | \(2177286259681/105875000\) | \(1205982421875000\) | \([2]\) | \(221184\) | \(1.6535\) | |
| 34650.dg3 | 34650cz3 | \([1, -1, 1, -20255, 9159747]\) | \(-80677568161/3131816380\) | \(-35673345953437500\) | \([2]\) | \(331776\) | \(1.8563\) | |
| 34650.dg1 | 34650cz4 | \([1, -1, 1, -792005, 270011247]\) | \(4823468134087681/30382271150\) | \(346073057317968750\) | \([2]\) | \(663552\) | \(2.2028\) |
Rank
sage: E.rank()
The elliptic curves in class 34650cz have rank \(0\).
Complex multiplication
The elliptic curves in class 34650cz do not have complex multiplication.Modular form 34650.2.a.cz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
