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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 203280.bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 203280.bj1 | 203280de8 | \([0, -1, 0, -10307265976, -402772335706640]\) | \(16689299266861680229173649/2396798250\) | \(17391920351511552000\) | \([2]\) | \(119439360\) | \(4.0145\) | |
| 203280.bj2 | 203280de7 | \([0, -1, 0, -661145976, -5944669658640]\) | \(4404531606962679693649/444872222400201750\) | \(3228132471552097535720448000\) | \([2]\) | \(119439360\) | \(4.0145\) | |
| 203280.bj3 | 203280de6 | \([0, -1, 0, -644205976, -6293118682640]\) | \(4074571110566294433649/48828650062500\) | \(354316010018294016000000\) | \([2, 2]\) | \(59719680\) | \(3.6679\) | |
| 203280.bj4 | 203280de4 | \([0, -1, 0, -145221336, 672303150576]\) | \(46676570542430835889/106752955783320\) | \(774632952219460249681920\) | \([2]\) | \(39813120\) | \(3.4652\) | |
| 203280.bj5 | 203280de5 | \([0, -1, 0, -127410136, -551003519504]\) | \(31522423139920199089/164434491947880\) | \(1193188282325722073825280\) | \([2]\) | \(39813120\) | \(3.4652\) | |
| 203280.bj6 | 203280de3 | \([0, -1, 0, -39205976, -103726682640]\) | \(-918468938249433649/109183593750000\) | \(-792270424176000000000000\) | \([2]\) | \(29859840\) | \(3.3214\) | |
| 203280.bj7 | 203280de2 | \([0, -1, 0, -12411736, 2092785136]\) | \(29141055407581489/16604321025600\) | \(120486164727533410713600\) | \([2, 2]\) | \(19906560\) | \(3.1186\) | |
| 203280.bj8 | 203280de1 | \([0, -1, 0, 3076264, 259005936]\) | \(443688652450511/260789760000\) | \(-1892372348990914560000\) | \([2]\) | \(9953280\) | \(2.7721\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 203280.bj have rank \(2\).
Complex multiplication
The elliptic curves in class 203280.bj do not have complex multiplication.Modular form 203280.2.a.bj
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
