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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 62475f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
62475.cd4 | 62475f1 | \([1, 1, 0, -221750, 91248375]\) | \(-656008386769/1581036975\) | \(-2906365922996484375\) | \([2]\) | \(884736\) | \(2.2309\) | \(\Gamma_0(N)\)-optimal |
62475.cd3 | 62475f2 | \([1, 1, 0, -4686875, 3900000000]\) | \(6193921595708449/6452105625\) | \(11860683979306640625\) | \([2, 2]\) | \(1769472\) | \(2.5775\) | |
62475.cd2 | 62475f3 | \([1, 1, 0, -5844500, 1824378375]\) | \(12010404962647729/6166198828125\) | \(11335111342657470703125\) | \([2]\) | \(3538944\) | \(2.9241\) | |
62475.cd1 | 62475f4 | \([1, 1, 0, -74971250, 249825028125]\) | \(25351269426118370449/27551475\) | \(50646929410546875\) | \([2]\) | \(3538944\) | \(2.9241\) |
Rank
sage: E.rank()
The elliptic curves in class 62475f have rank \(0\).
Complex multiplication
The elliptic curves in class 62475f do not have complex multiplication.Modular form 62475.2.a.f
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.