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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 3525.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3525.j1 | 3525g4 | \([1, 1, 0, -18800, 984375]\) | \(47034153084673/141\) | \(2203125\) | \([2]\) | \(3072\) | \(0.87007\) | |
3525.j2 | 3525g3 | \([1, 1, 0, -1550, 4125]\) | \(26383748833/14639043\) | \(228735046875\) | \([2]\) | \(3072\) | \(0.87007\) | |
3525.j3 | 3525g2 | \([1, 1, 0, -1175, 15000]\) | \(11497268593/19881\) | \(310640625\) | \([2, 2]\) | \(1536\) | \(0.52350\) | |
3525.j4 | 3525g1 | \([1, 1, 0, -50, 375]\) | \(-912673/3807\) | \(-59484375\) | \([2]\) | \(768\) | \(0.17693\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3525.j have rank \(0\).
Complex multiplication
The elliptic curves in class 3525.j do not have complex multiplication.Modular form 3525.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.