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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 21780.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21780.r1 | 21780x4 | \([0, 0, 0, -272246007, -1728981679106]\) | \(6749703004355978704/5671875\) | \(1875211490988000000\) | \([2]\) | \(1658880\) | \(3.2418\) | |
21780.r2 | 21780x3 | \([0, 0, 0, -17011632, -27027819731]\) | \(-26348629355659264/24169921875\) | \(-499434878636718750000\) | \([2]\) | \(829440\) | \(2.8952\) | |
21780.r3 | 21780x2 | \([0, 0, 0, -3437247, -2258565914]\) | \(13584145739344/1195803675\) | \(395351588729596435200\) | \([2]\) | \(552960\) | \(2.6925\) | |
21780.r4 | 21780x1 | \([0, 0, 0, 238128, -164337239]\) | \(72268906496/606436875\) | \(-12531100788527310000\) | \([2]\) | \(276480\) | \(2.3459\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21780.r have rank \(1\).
Complex multiplication
The elliptic curves in class 21780.r do not have complex multiplication.Modular form 21780.2.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 LMFDB 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 LMFDB labels.