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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 4032f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4032.be3 | 4032f1 | \([0, 0, 0, -264, 1640]\) | \(2725888/21\) | \(15676416\) | \([2]\) | \(1024\) | \(0.20969\) | \(\Gamma_0(N)\)-optimal |
4032.be2 | 4032f2 | \([0, 0, 0, -444, -880]\) | \(810448/441\) | \(5267275776\) | \([2, 2]\) | \(2048\) | \(0.55626\) | |
4032.be1 | 4032f3 | \([0, 0, 0, -5484, -156112]\) | \(381775972/567\) | \(27088846848\) | \([2]\) | \(4096\) | \(0.90284\) | |
4032.be4 | 4032f4 | \([0, 0, 0, 1716, -6928]\) | \(11696828/7203\) | \(-344128684032\) | \([2]\) | \(4096\) | \(0.90284\) |
Rank
sage: E.rank()
The elliptic curves in class 4032f have rank \(0\).
Complex multiplication
The elliptic curves in class 4032f do not have complex multiplication.Modular form 4032.2.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.