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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 137904bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137904.bh5 | 137904bu1 | \([0, -1, 0, -91992, -9929232]\) | \(4354703137/352512\) | \(6969377153875968\) | \([2]\) | \(884736\) | \(1.7830\) | \(\Gamma_0(N)\)-optimal |
| 137904.bh4 | 137904bu2 | \([0, -1, 0, -308312, 54447600]\) | \(163936758817/30338064\) | \(599802021305450496\) | \([2, 2]\) | \(1769472\) | \(2.1296\) | |
| 137904.bh2 | 137904bu3 | \([0, -1, 0, -4688792, 3909270000]\) | \(576615941610337/27060804\) | \(535008593078009856\) | \([2, 2]\) | \(3538944\) | \(2.4761\) | |
| 137904.bh6 | 137904bu4 | \([0, -1, 0, 611048, 316281328]\) | \(1276229915423/2927177028\) | \(-57872074438015598592\) | \([2]\) | \(3538944\) | \(2.4761\) | |
| 137904.bh1 | 137904bu5 | \([0, -1, 0, -75019832, 250124174832]\) | \(2361739090258884097/5202\) | \(102846711472128\) | \([2]\) | \(7077888\) | \(2.8227\) | |
| 137904.bh3 | 137904bu6 | \([0, -1, 0, -4445432, 4332911088]\) | \(-491411892194497/125563633938\) | \(-2482469594581581176832\) | \([2]\) | \(7077888\) | \(2.8227\) |
Rank
sage: E.rank()
The elliptic curves in class 137904bu have rank \(1\).
Complex multiplication
The elliptic curves in class 137904bu do not have complex multiplication.Modular form 137904.2.a.bu
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
