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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 269610.bh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 269610.bh1 | 269610bh8 | \([1, 0, 0, -255945779406, -49839107663440980]\) | \(1854247188432036912132702899859171123169/6226499397577271484375000\) | \(6226499397577271484375000\) | \([2]\) | \(1266057216\) | \(4.8627\) | |
| 269610.bh2 | 269610bh7 | \([1, 0, 0, -16922798686, -683505295776484]\) | \(535969746930822235944971662061093089/108346971031222480727652383913000\) | \(108346971031222480727652383913000\) | \([2]\) | \(1266057216\) | \(4.8627\) | |
| 269610.bh3 | 269610bh6 | \([1, 0, 0, -15996833686, -778714313427484]\) | \(452715955427640171093925801003733089/26232835616568562566321000000\) | \(26232835616568562566321000000\) | \([2, 2]\) | \(633028608\) | \(4.5162\) | |
| 269610.bh4 | 269610bh4 | \([1, 0, 0, -5174274646, 143175689502500]\) | \(15320471914877680149062673942469729/10272375892442530969162283520\) | \(10272375892442530969162283520\) | \([6]\) | \(422019072\) | \(4.3134\) | |
| 269610.bh5 | 269610bh5 | \([1, 0, 0, -3171182166, -67850404035804]\) | \(3526852390105847556279202024922209/52229426872675156208533440000\) | \(52229426872675156208533440000\) | \([6]\) | \(422019072\) | \(4.3134\) | |
| 269610.bh6 | 269610bh3 | \([1, 0, 0, -942151766, -13632367316700]\) | \(-92488196675669204123085195328609/26759314786942668288112128000\) | \(-26759314786942668288112128000\) | \([2]\) | \(316514304\) | \(4.1696\) | |
| 269610.bh7 | 269610bh2 | \([1, 0, 0, -387369046, 1288935375140]\) | \(6428334307711763111670164831329/3002359821568108533389721600\) | \(3002359821568108533389721600\) | \([2, 6]\) | \(211009536\) | \(3.9669\) | |
| 269610.bh8 | 269610bh1 | \([1, 0, 0, 85800874, 152286593316]\) | \(69855217159487253970629608351/50443133349703832221777920\) | \(-50443133349703832221777920\) | \([6]\) | \(105504768\) | \(3.6203\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 269610.bh have rank \(0\).
Complex multiplication
The elliptic curves in class 269610.bh do not have complex multiplication.Modular form 269610.2.a.bh
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.
