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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 33150cg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33150.cl3 | 33150cg1 | \([1, 0, 0, -14688, 682992]\) | \(22428153804601/35802000\) | \(559406250000\) | \([4]\) | \(129024\) | \(1.1518\) | \(\Gamma_0(N)\)-optimal |
33150.cl2 | 33150cg2 | \([1, 0, 0, -19188, 228492]\) | \(50002789171321/27473062500\) | \(429266601562500\) | \([2, 2]\) | \(258048\) | \(1.4984\) | |
33150.cl4 | 33150cg3 | \([1, 0, 0, 74562, 1822242]\) | \(2933972022568679/1789082460750\) | \(-27954413449218750\) | \([2]\) | \(516096\) | \(1.8450\) | |
33150.cl1 | 33150cg4 | \([1, 0, 0, -184938, -30435258]\) | \(44769506062996441/323730468750\) | \(5058288574218750\) | \([2]\) | \(516096\) | \(1.8450\) |
Rank
sage: E.rank()
The elliptic curves in class 33150cg have rank \(1\).
Complex multiplication
The elliptic curves in class 33150cg do not have complex multiplication.Modular form 33150.2.a.cg
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.