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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 29370.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29370.bm1 | 29370bp4 | \([1, 0, 0, -1064189920, -12637949352220]\) | \(133284956652244710243152075681281/8135392425834393812901934620\) | \(8135392425834393812901934620\) | \([2]\) | \(25600000\) | \(4.1076\) | |
| 29370.bm2 | 29370bp3 | \([1, 0, 0, -1048084820, -13060028592000]\) | \(127324800640445734294052812418881/521410035333380777456400\) | \(521410035333380777456400\) | \([2]\) | \(12800000\) | \(3.7610\) | |
| 29370.bm3 | 29370bp2 | \([1, 0, 0, -187842820, 990906448400]\) | \(733005968209216932163418210881/972182437259299200000\) | \(972182437259299200000\) | \([10]\) | \(5120000\) | \(3.3029\) | |
| 29370.bm4 | 29370bp1 | \([1, 0, 0, -11842820, 15197648400]\) | \(183691516586815867994210881/6511602493440000000000\) | \(6511602493440000000000\) | \([10]\) | \(2560000\) | \(2.9563\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 29370.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 29370.bm do not have complex multiplication.Modular form 29370.2.a.bm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
