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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 258570fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
258570.fd2 | 258570fd1 | \([1, -1, 1, -509567, -108116121]\) | \(1892819053/440640\) | \(3406446854110946880\) | \([2]\) | \(5271552\) | \(2.2680\) | \(\Gamma_0(N)\)-optimal |
258570.fd1 | 258570fd2 | \([1, -1, 1, -7627847, -8106215529]\) | \(6349095794413/520200\) | \(4021499758325423400\) | \([2]\) | \(10543104\) | \(2.6146\) |
Rank
sage: E.rank()
The elliptic curves in class 258570fd have rank \(1\).
Complex multiplication
The elliptic curves in class 258570fd do not have complex multiplication.Modular form 258570.2.a.fd
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.