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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 57330r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
57330.r3 | 57330r1 | \([1, -1, 0, -2655, 338701]\) | \(-24137569/561600\) | \(-48166253553600\) | \([2]\) | \(138240\) | \(1.3062\) | \(\Gamma_0(N)\)-optimal |
57330.r2 | 57330r2 | \([1, -1, 0, -90855, 10516981]\) | \(967068262369/4928040\) | \(422658874932840\) | \([2]\) | \(276480\) | \(1.6528\) | |
57330.r4 | 57330r3 | \([1, -1, 0, 23805, -8938175]\) | \(17394111071/411937500\) | \(-35330281469437500\) | \([2]\) | \(414720\) | \(1.8555\) | |
57330.r1 | 57330r4 | \([1, -1, 0, -527445, -139804925]\) | \(189208196468929/10860320250\) | \(931447540660250250\) | \([2]\) | \(829440\) | \(2.2021\) |
Rank
sage: E.rank()
The elliptic curves in class 57330r have rank \(1\).
Complex multiplication
The elliptic curves in class 57330r do not have complex multiplication.Modular form 57330.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.