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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 438.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
438.c1 | 438d4 | \([1, 0, 1, -32681, 1883180]\) | \(3860029467400479625/697348114739712\) | \(697348114739712\) | \([2]\) | \(3456\) | \(1.5671\) | |
438.c2 | 438d2 | \([1, 0, 1, -31106, 2108972]\) | \(3328404840479049625/31078728\) | \(31078728\) | \([6]\) | \(1152\) | \(1.0178\) | |
438.c3 | 438d3 | \([1, 0, 1, -9641, -337876]\) | \(99088945018143625/8260256268288\) | \(8260256268288\) | \([2]\) | \(1728\) | \(1.2206\) | |
438.c4 | 438d1 | \([1, 0, 1, -1946, 32780]\) | \(814388006841625/2482892352\) | \(2482892352\) | \([6]\) | \(576\) | \(0.67126\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 438.c have rank \(1\).
Complex multiplication
The elliptic curves in class 438.c do not have complex multiplication.Modular form 438.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.