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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 303450.bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.bb1 | 303450bb6 | \([1, 1, 0, -121380150, -514768889250]\) | \(524388516989299201/3150\) | \(1188020974218750\) | \([2]\) | \(31457280\) | \(2.9581\) | |
| 303450.bb2 | 303450bb4 | \([1, 1, 0, -7586400, -8045320500]\) | \(128031684631201/9922500\) | \(3742266068789062500\) | \([2, 2]\) | \(15728640\) | \(2.6115\) | |
| 303450.bb3 | 303450bb5 | \([1, 1, 0, -7080650, -9163533750]\) | \(-104094944089921/35880468750\) | \(-13532301409460449218750\) | \([2]\) | \(31457280\) | \(2.9581\) | |
| 303450.bb4 | 303450bb3 | \([1, 1, 0, -2673400, 1589072500]\) | \(5602762882081/345888060\) | \(130451514289470937500\) | \([2]\) | \(15728640\) | \(2.6115\) | |
| 303450.bb5 | 303450bb2 | \([1, 1, 0, -505900, -108080000]\) | \(37966934881/8643600\) | \(3259929553256250000\) | \([2, 2]\) | \(7864320\) | \(2.2649\) | |
| 303450.bb6 | 303450bb1 | \([1, 1, 0, 72100, -10398000]\) | \(109902239/188160\) | \(-70964452860000000\) | \([2]\) | \(3932160\) | \(1.9184\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 303450.bb have rank \(2\).
Complex multiplication
The elliptic curves in class 303450.bb do not have complex multiplication.Modular form 303450.2.a.bb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
