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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 225318.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 225318.bg1 | 225318e5 | \([1, 0, 0, -61286542, 184664655122]\) | \(2361739090258884097/5202\) | \(56073478141458\) | \([2]\) | \(13565952\) | \(2.7722\) | |
| 225318.bg2 | 225318e3 | \([1, 0, 0, -3830452, 2885077580]\) | \(576615941610337/27060804\) | \(291694233291864516\) | \([2, 2]\) | \(6782976\) | \(2.4256\) | |
| 225318.bg3 | 225318e6 | \([1, 0, 0, -3631642, 3197964758]\) | \(-491411892194497/125563633938\) | \(-1353477447709434235602\) | \([2]\) | \(13565952\) | \(2.7722\) | |
| 225318.bg4 | 225318e2 | \([1, 0, 0, -251872, 40106480]\) | \(163936758817/30338064\) | \(327020524520983056\) | \([2, 2]\) | \(3391488\) | \(2.0790\) | |
| 225318.bg5 | 225318e1 | \([1, 0, 0, -75152, -7360512]\) | \(4354703137/352512\) | \(3799802754056448\) | \([2]\) | \(1695744\) | \(1.7324\) | \(\Gamma_0(N)\)-optimal |
| 225318.bg6 | 225318e4 | \([1, 0, 0, 499188, 233729748]\) | \(1276229915423/2927177028\) | \(-31552671490914262212\) | \([2]\) | \(6782976\) | \(2.4256\) |
Rank
sage: E.rank()
The elliptic curves in class 225318.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 225318.bg do not have complex multiplication.Modular form 225318.2.a.bg
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.
