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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 180336.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 180336.bg1 | 180336bp5 | \([0, -1, 0, -93284672, 346765316928]\) | \(908031902324522977/161726530797\) | \(15989494973412198371328\) | \([2]\) | \(18874368\) | \(3.2649\) | |
| 180336.bg2 | 180336bp3 | \([0, -1, 0, -6422832, 4251709440]\) | \(296380748763217/92608836489\) | \(9156002532405064667136\) | \([2, 2]\) | \(9437184\) | \(2.9183\) | |
| 180336.bg3 | 180336bp2 | \([0, -1, 0, -2515552, -1484177600]\) | \(17806161424897/668584449\) | \(66101261394184114176\) | \([2, 2]\) | \(4718592\) | \(2.5718\) | |
| 180336.bg4 | 180336bp1 | \([0, -1, 0, -2492432, -1513715712]\) | \(17319700013617/25857\) | \(2556416498208768\) | \([2]\) | \(2359296\) | \(2.2252\) | \(\Gamma_0(N)\)-optimal |
| 180336.bg5 | 180336bp4 | \([0, -1, 0, 1021808, -5329995392]\) | \(1193377118543/124806800313\) | \(-12339334161302565261312\) | \([4]\) | \(9437184\) | \(2.9183\) | |
| 180336.bg6 | 180336bp6 | \([0, -1, 0, 17922528, 28772356032]\) | \(6439735268725823/7345472585373\) | \(-726228383199433434058752\) | \([2]\) | \(18874368\) | \(3.2649\) |
Rank
sage: E.rank()
The elliptic curves in class 180336.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 180336.bg do not have complex multiplication.Modular form 180336.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
