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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 2850.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2850.p1 | 2850q3 | \([1, 1, 1, -30836088, -65920459719]\) | \(207530301091125281552569/805586668007040\) | \(12587291687610000000\) | \([2]\) | \(215040\) | \(2.8781\) | |
2850.p2 | 2850q4 | \([1, 1, 1, -5844088, 4196148281]\) | \(1412712966892699019449/330160465517040000\) | \(5158757273703750000000\) | \([2]\) | \(215040\) | \(2.8781\) | |
2850.p3 | 2850q2 | \([1, 1, 1, -1956088, -998219719]\) | \(52974743974734147769/3152005008998400\) | \(49250078265600000000\) | \([2, 2]\) | \(107520\) | \(2.5316\) | |
2850.p4 | 2850q1 | \([1, 1, 1, 91912, -64331719]\) | \(5495662324535111/117739817533440\) | \(-1839684648960000000\) | \([4]\) | \(53760\) | \(2.1850\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2850.p have rank \(0\).
Complex multiplication
The elliptic curves in class 2850.p do not have complex multiplication.Modular form 2850.2.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.