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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 86640ct
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86640.y4 | 86640ct1 | \([0, -1, 0, 129840, -81446400]\) | \(1256216039/15582375\) | \(-3002722549544448000\) | \([2]\) | \(1658880\) | \(2.2264\) | \(\Gamma_0(N)\)-optimal |
86640.y3 | 86640ct2 | \([0, -1, 0, -2209440, -1179972288]\) | \(6189976379881/456890625\) | \(88042790804544000000\) | \([2, 2]\) | \(3317760\) | \(2.5729\) | |
86640.y2 | 86640ct3 | \([0, -1, 0, -7147920, 5959094400]\) | \(209595169258201/41748046875\) | \(8044845651000000000000\) | \([4]\) | \(6635520\) | \(2.9195\) | |
86640.y1 | 86640ct4 | \([0, -1, 0, -34699440, -78662124288]\) | \(23977812996389881/146611125\) | \(28251953315947008000\) | \([2]\) | \(6635520\) | \(2.9195\) |
Rank
sage: E.rank()
The elliptic curves in class 86640ct have rank \(0\).
Complex multiplication
The elliptic curves in class 86640ct do not have complex multiplication.Modular form 86640.2.a.ct
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.