Show commands:
SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 115710.bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
115710.bl1 | 115710bk8 | \([1, 0, 1, -16061619848, -783488938704514]\) | \(458238674512589140162512851063637241/392468692344120\) | \(392468692344120\) | \([2]\) | \(79626240\) | \(3.9829\) | |
115710.bl2 | 115710bk7 | \([1, 0, 1, -1005154648, -12208693503874]\) | \(112311018900728619777866774440441/605137023657856954297300680\) | \(605137023657856954297300680\) | \([2]\) | \(79626240\) | \(3.9829\) | |
115710.bl3 | 115710bk6 | \([1, 0, 1, -1003851248, -12242077227394]\) | \(111874678752008055543640205694841/3274073546848947662400\) | \(3274073546848947662400\) | \([2, 2]\) | \(39813120\) | \(3.6363\) | |
115710.bl4 | 115710bk5 | \([1, 0, 1, -198296123, -1074708019744]\) | \(862315869068763192029111676841/81885458731960991976750\) | \(81885458731960991976750\) | \([6]\) | \(26542080\) | \(3.4336\) | |
115710.bl5 | 115710bk4 | \([1, 0, 1, -73328623, 229914901256]\) | \(43605803425346349400425156841/2397733750298041341023250\) | \(2397733750298041341023250\) | \([6]\) | \(26542080\) | \(3.4336\) | |
115710.bl6 | 115710bk3 | \([1, 0, 1, -62659248, -191807812994]\) | \(-27206911093874365984986366841/147776899215941936640000\) | \(-147776899215941936640000\) | \([2]\) | \(19906560\) | \(3.2898\) | |
115710.bl7 | 115710bk2 | \([1, 0, 1, -13312373, -14159184244]\) | \(260909623523179393528416841/64394497051541618062500\) | \(64394497051541618062500\) | \([2, 6]\) | \(13271040\) | \(3.0870\) | |
115710.bl8 | 115710bk1 | \([1, 0, 1, 2000127, -1400809244]\) | \(884905188895571476583159/1359998795777343750000\) | \(-1359998795777343750000\) | \([6]\) | \(6635520\) | \(2.7405\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 115710.bl have rank \(1\).
Complex multiplication
The elliptic curves in class 115710.bl do not have complex multiplication.Modular form 115710.2.a.bl
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.