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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 37026.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37026.bm1 | 37026bj6 | \([1, -1, 1, -30213239, -63913516675]\) | \(2361739090258884097/5202\) | \(6718216374738\) | \([2]\) | \(1310720\) | \(2.5953\) | |
| 37026.bm2 | 37026bj4 | \([1, -1, 1, -1888349, -998271007]\) | \(576615941610337/27060804\) | \(34948161581387076\) | \([2, 2]\) | \(655360\) | \(2.2488\) | |
| 37026.bm3 | 37026bj5 | \([1, -1, 1, -1790339, -1106591659]\) | \(-491411892194497/125563633938\) | \(-162161411302168331922\) | \([2]\) | \(1310720\) | \(2.5953\) | |
| 37026.bm4 | 37026bj2 | \([1, -1, 1, -124169, -13858567]\) | \(163936758817/30338064\) | \(39180637897472016\) | \([2, 2]\) | \(327680\) | \(1.9022\) | |
| 37026.bm5 | 37026bj1 | \([1, -1, 1, -37049, 2554841]\) | \(4354703137/352512\) | \(455257956688128\) | \([2]\) | \(163840\) | \(1.5556\) | \(\Gamma_0(N)\)-optimal |
| 37026.bm6 | 37026bj3 | \([1, -1, 1, 246091, -80949679]\) | \(1276229915423/2927177028\) | \(-3780355371254616132\) | \([2]\) | \(655360\) | \(2.2488\) |
Rank
sage: E.rank()
The elliptic curves in class 37026.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 37026.bm do not have complex multiplication.Modular form 37026.2.a.bm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
