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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 84270.bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84270.bb1 | 84270bg7 | \([1, 1, 1, -14981860, -22326372163]\) | \(16778985534208729/81000\) | \(1795313251449000\) | \([2]\) | \(3594240\) | \(2.5493\) | |
| 84270.bb2 | 84270bg8 | \([1, 1, 1, -1273940, -75856195]\) | \(10316097499609/5859375000\) | \(129869303490234375000\) | \([2]\) | \(3594240\) | \(2.5493\) | |
| 84270.bb3 | 84270bg6 | \([1, 1, 1, -936860, -348756163]\) | \(4102915888729/9000000\) | \(199479250161000000\) | \([2, 2]\) | \(1797120\) | \(2.2027\) | |
| 84270.bb4 | 84270bg5 | \([1, 1, 1, -810455, 280487927]\) | \(2656166199049/33750\) | \(748047188103750\) | \([2]\) | \(1198080\) | \(2.0000\) | |
| 84270.bb5 | 84270bg4 | \([1, 1, 1, -192475, -28075105]\) | \(35578826569/5314410\) | \(117790502427568890\) | \([2]\) | \(1198080\) | \(2.0000\) | |
| 84270.bb6 | 84270bg2 | \([1, 1, 1, -52025, 4116035]\) | \(702595369/72900\) | \(1615781926304100\) | \([2, 2]\) | \(599040\) | \(1.6534\) | |
| 84270.bb7 | 84270bg3 | \([1, 1, 1, -37980, -9339075]\) | \(-273359449/1536000\) | \(-34044458694144000\) | \([2]\) | \(898560\) | \(1.8562\) | |
| 84270.bb8 | 84270bg1 | \([1, 1, 1, 4155, 318267]\) | \(357911/2160\) | \(-47875020038640\) | \([2]\) | \(299520\) | \(1.3069\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84270.bb have rank \(1\).
Complex multiplication
The elliptic curves in class 84270.bb do not have complex multiplication.Modular form 84270.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
