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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 164730.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
164730.p1 | 164730cl4 | \([1, 1, 0, -356465177, -2590581283611]\) | \(207530301091125281552569/805586668007040\) | \(19444903784500020485760\) | \([2]\) | \(45875200\) | \(3.4900\) | |
164730.p2 | 164730cl3 | \([1, 1, 0, -67557657, 164992969701]\) | \(1412712966892699019449/330160465517040000\) | \(7969271017489673675760000\) | \([2]\) | \(45875200\) | \(3.4900\) | |
164730.p3 | 164730cl2 | \([1, 1, 0, -22612377, -39211415451]\) | \(52974743974734147769/3152005008998400\) | \(76081738393044500889600\) | \([2, 2]\) | \(22937600\) | \(3.1434\) | |
164730.p4 | 164730cl1 | \([1, 1, 0, 1062503, -2529556379]\) | \(5495662324535111/117739817533440\) | \(-2841952969760817807360\) | \([2]\) | \(11468800\) | \(2.7969\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 164730.p have rank \(0\).
Complex multiplication
The elliptic curves in class 164730.p do not have complex multiplication.Modular form 164730.2.a.p
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.