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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 162288.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162288.bg1 | 162288o5 | \([0, 0, 0, -558369651, 5078394744626]\) | \(54804145548726848737/637608031452\) | \(223990446391640218386432\) | \([2]\) | \(37748736\) | \(3.6314\) | |
| 162288.bg2 | 162288o4 | \([0, 0, 0, -124990131, -537836216974]\) | \(614716917569296417/19093020912\) | \(6707340663988725153792\) | \([2]\) | \(18874368\) | \(3.2848\) | |
| 162288.bg3 | 162288o3 | \([0, 0, 0, -35802291, 75021299570]\) | \(14447092394873377/1439452851984\) | \(505676953506016600326144\) | \([2, 2]\) | \(18874368\) | \(3.2848\) | |
| 162288.bg4 | 162288o2 | \([0, 0, 0, -8142771, -7653005710]\) | \(169967019783457/26337394944\) | \(9252275001720390549504\) | \([2, 2]\) | \(9437184\) | \(2.9382\) | |
| 162288.bg5 | 162288o1 | \([0, 0, 0, 888909, -660679054]\) | \(221115865823/664731648\) | \(-233518919495259783168\) | \([2]\) | \(4718592\) | \(2.5917\) | \(\Gamma_0(N)\)-optimal |
| 162288.bg6 | 162288o6 | \([0, 0, 0, 44212749, 362803392434]\) | \(27207619911317663/177609314617308\) | \(-62393802621727156724809728\) | \([2]\) | \(37748736\) | \(3.6314\) |
Rank
sage: E.rank()
The elliptic curves in class 162288.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 162288.bg do not have complex multiplication.Modular form 162288.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
