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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 336336.gn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 336336.gn1 | 336336gn4 | \([0, 1, 0, -51097608, -140590669836]\) | \(30618029936661765625/3678951124992\) | \(1772850876023536877568\) | \([2]\) | \(23887872\) | \(3.1019\) | |
| 336336.gn2 | 336336gn3 | \([0, 1, 0, -2928648, -2576965644]\) | \(-5764706497797625/2612665516032\) | \(-1259018179770977353728\) | \([2]\) | \(11943936\) | \(2.7554\) | |
| 336336.gn3 | 336336gn2 | \([0, 1, 0, -1411608, 367319700]\) | \(645532578015625/252306960048\) | \(121584277678846574592\) | \([2]\) | \(7962624\) | \(2.5526\) | |
| 336336.gn4 | 336336gn1 | \([0, 1, 0, 281832, 41501844]\) | \(5137417856375/4510142208\) | \(-2173393799696351232\) | \([2]\) | \(3981312\) | \(2.2060\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 336336.gn have rank \(1\).
Complex multiplication
The elliptic curves in class 336336.gn do not have complex multiplication.Modular form 336336.2.a.gn
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.
