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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 69312.ct
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69312.ct1 | 69312bo3 | \([0, 1, 0, -9888993, 11966111871]\) | \(8671983378625/82308\) | \(1015087726158938112\) | \([2]\) | \(2488320\) | \(2.6174\) | |
| 69312.ct2 | 69312bo4 | \([0, 1, 0, -9657953, 12552075519]\) | \(-8078253774625/846825858\) | \(-10443730070586234765312\) | \([2]\) | \(4976640\) | \(2.9640\) | |
| 69312.ct3 | 69312bo1 | \([0, 1, 0, -185313, -2407041]\) | \(57066625/32832\) | \(404910339520462848\) | \([2]\) | \(829440\) | \(2.0681\) | \(\Gamma_0(N)\)-optimal |
| 69312.ct4 | 69312bo2 | \([0, 1, 0, 738847, -18487425]\) | \(3616805375/2105352\) | \(-25964875521749680128\) | \([2]\) | \(1658880\) | \(2.4147\) |
Rank
sage: E.rank()
The elliptic curves in class 69312.ct have rank \(1\).
Complex multiplication
The elliptic curves in class 69312.ct do not have complex multiplication.Modular form 69312.2.a.ct
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
