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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 433200p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.p4 | 433200p1 | \([0, -1, 0, 1801992, -812185488]\) | \(214921799/218880\) | \(-659033755729920000000\) | \([2]\) | \(26542080\) | \(2.6816\) | \(\Gamma_0(N)\)-optimal* |
433200.p3 | 433200p2 | \([0, -1, 0, -9750008, -7466137488]\) | \(34043726521/11696400\) | \(35217116321817600000000\) | \([2, 2]\) | \(53084160\) | \(3.0282\) | \(\Gamma_0(N)\)-optimal* |
433200.p2 | 433200p3 | \([0, -1, 0, -64622008, 194462822512]\) | \(9912050027641/311647500\) | \(938350796732640000000000\) | \([2]\) | \(106168320\) | \(3.3748\) | \(\Gamma_0(N)\)-optimal* |
433200.p1 | 433200p4 | \([0, -1, 0, -139710008, -635432857488]\) | \(100162392144121/23457780\) | \(70629883289867520000000\) | \([2]\) | \(106168320\) | \(3.3748\) |
Rank
sage: E.rank()
The elliptic curves in class 433200p have rank \(1\).
Complex multiplication
The elliptic curves in class 433200p do not have complex multiplication.Modular form 433200.2.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 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.