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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 6270.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6270.r1 | 6270r4 | \([1, 0, 0, -4895705, -4128462873]\) | \(12976854634417729473922321/148112152782766327650\) | \(148112152782766327650\) | \([2]\) | \(320000\) | \(2.6836\) | |
| 6270.r2 | 6270r2 | \([1, 0, 0, -454955, 118072977]\) | \(10414276373665867414321/301547812500000\) | \(301547812500000\) | \([10]\) | \(64000\) | \(1.8788\) | |
| 6270.r3 | 6270r3 | \([1, 0, 0, -64175, -163709355]\) | \(-29229525625065721201/11560253601080069820\) | \(-11560253601080069820\) | \([2]\) | \(160000\) | \(2.3370\) | |
| 6270.r4 | 6270r1 | \([1, 0, 0, -27275, 2000625]\) | \(-2243980016705847601/434411683200000\) | \(-434411683200000\) | \([10]\) | \(32000\) | \(1.5323\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6270.r have rank \(1\).
Complex multiplication
The elliptic curves in class 6270.r do not have complex multiplication.Modular form 6270.2.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.
\(\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
