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SageMath
E = EllipticCurve("bs1")
E.isogeny_class()
Elliptic curves in class 177870.bs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.bs1 | 177870hp7 | \([1, 1, 0, -38248102, 57264612574]\) | \(29689921233686449/10380965400750\) | \(2163625516445874705666750\) | \([2]\) | \(39813120\) | \(3.3709\) | |
| 177870.bs2 | 177870hp4 | \([1, 1, 0, -34157092, 76822593016]\) | \(21145699168383889/2593080\) | \(540455905361184120\) | \([2]\) | \(13271040\) | \(2.8216\) | |
| 177870.bs3 | 177870hp6 | \([1, 1, 0, -16014352, -24017530676]\) | \(2179252305146449/66177562500\) | \(13792885084738553062500\) | \([2, 2]\) | \(19906560\) | \(3.0244\) | |
| 177870.bs4 | 177870hp3 | \([1, 1, 0, -15895772, -24399903744]\) | \(2131200347946769/2058000\) | \(428933258223162000\) | \([2]\) | \(9953280\) | \(2.6778\) | |
| 177870.bs5 | 177870hp2 | \([1, 1, 0, -2140492, 1192980496]\) | \(5203798902289/57153600\) | \(11912089342654670400\) | \([2, 2]\) | \(6635520\) | \(2.4750\) | |
| 177870.bs6 | 177870hp5 | \([1, 1, 0, -480372, 2998194984]\) | \(-58818484369/18600435000\) | \(-3876746933390738715000\) | \([2]\) | \(13271040\) | \(2.8216\) | |
| 177870.bs7 | 177870hp1 | \([1, 1, 0, -243212, -16345776]\) | \(7633736209/3870720\) | \(806744675058094080\) | \([2]\) | \(3317760\) | \(2.1285\) | \(\Gamma_0(N)\)-optimal |
| 177870.bs8 | 177870hp8 | \([1, 1, 0, 4322118, -80825425974]\) | \(42841933504271/13565917968750\) | \(-2827440911139007324218750\) | \([2]\) | \(39813120\) | \(3.3709\) |
Rank
sage: E.rank()
The elliptic curves in class 177870.bs have rank \(0\).
Complex multiplication
The elliptic curves in class 177870.bs do not have complex multiplication.Modular form 177870.2.a.bs
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
