Show commands:
SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 8670bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8670.bb3 | 8670bb1 | \([1, 0, 0, -29195, 1674225]\) | \(114013572049/15667200\) | \(378168121036800\) | \([4]\) | \(55296\) | \(1.5240\) | \(\Gamma_0(N)\)-optimal |
| 8670.bb2 | 8670bb2 | \([1, 0, 0, -121675, -14657743]\) | \(8253429989329/936360000\) | \(22601454108840000\) | \([2, 2]\) | \(110592\) | \(1.8706\) | |
| 8670.bb1 | 8670bb3 | \([1, 0, 0, -1890355, -1000519975]\) | \(30949975477232209/478125000\) | \(11540775178125000\) | \([2]\) | \(221184\) | \(2.2171\) | |
| 8670.bb4 | 8670bb4 | \([1, 0, 0, 167325, -73671543]\) | \(21464092074671/109596256200\) | \(-2645387196169177800\) | \([2]\) | \(221184\) | \(2.2171\) |
Rank
sage: E.rank()
The elliptic curves in class 8670bb have rank \(0\).
Complex multiplication
The elliptic curves in class 8670bb do not have complex multiplication.Modular form 8670.2.a.bb
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
