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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 6006g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6006.h3 | 6006g1 | \([1, 1, 0, -55764, 5045328]\) | \(19177749277229260873/10959556608\) | \(10959556608\) | \([2]\) | \(15360\) | \(1.2521\) | \(\Gamma_0(N)\)-optimal |
6006.h2 | 6006g2 | \([1, 1, 0, -56084, 4984080]\) | \(19509797127983109193/458190464187456\) | \(458190464187456\) | \([2, 2]\) | \(30720\) | \(1.5987\) | |
6006.h1 | 6006g3 | \([1, 1, 0, -124124, -9562872]\) | \(211493228575739333833/83312312835279528\) | \(83312312835279528\) | \([2]\) | \(61440\) | \(1.9453\) | |
6006.h4 | 6006g4 | \([1, 1, 0, 6836, 15617560]\) | \(35320805896348727/105350345460143784\) | \(-105350345460143784\) | \([2]\) | \(61440\) | \(1.9453\) |
Rank
sage: E.rank()
The elliptic curves in class 6006g have rank \(1\).
Complex multiplication
The elliptic curves in class 6006g do not have complex multiplication.Modular form 6006.2.a.g
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.