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SageMath
E = EllipticCurve("gj1")
E.isogeny_class()
Elliptic curves in class 33600.gj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.gj1 | 33600cq4 | \([0, 1, 0, -1412033, -646295937]\) | \(608119035935048/826875\) | \(423360000000000\) | \([2]\) | \(294912\) | \(2.0808\) | |
33600.gj2 | 33600cq3 | \([0, 1, 0, -224033, 27192063]\) | \(2428799546888/778248135\) | \(398463045120000000\) | \([2]\) | \(294912\) | \(2.0808\) | |
33600.gj3 | 33600cq2 | \([0, 1, 0, -89033, -9932937]\) | \(1219555693504/43758225\) | \(2800526400000000\) | \([2, 2]\) | \(147456\) | \(1.7343\) | |
33600.gj4 | 33600cq1 | \([0, 1, 0, 2092, -547062]\) | \(1012048064/130203045\) | \(-130203045000000\) | \([2]\) | \(73728\) | \(1.3877\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33600.gj have rank \(1\).
Complex multiplication
The elliptic curves in class 33600.gj do not have complex multiplication.Modular form 33600.2.a.gj
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.