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SageMath
E = EllipticCurve("gw1")
E.isogeny_class()
Elliptic curves in class 242550gw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 242550.gw6 | 242550gw1 | \([1, -1, 0, -3159032067, 68341447936341]\) | \(2601656892010848045529/56330588160\) | \(75488375627058240000000\) | \([2]\) | \(127401984\) | \(3.9168\) | \(\Gamma_0(N)\)-optimal |
| 242550.gw5 | 242550gw2 | \([1, -1, 0, -3162560067, 68181153256341]\) | \(2610383204210122997209/12104550027662400\) | \(16221254723797605399225000000\) | \([2, 2]\) | \(254803968\) | \(4.2633\) | |
| 242550.gw4 | 242550gw3 | \([1, -1, 0, -3370877442, 58652949757716]\) | \(3160944030998056790089/720291785342976000\) | \(965259881516120408064000000000\) | \([2]\) | \(382205952\) | \(4.4661\) | |
| 242550.gw7 | 242550gw4 | \([1, -1, 0, -1555115067, 137434706191341]\) | \(-310366976336070130009/5909282337130963560\) | \(-7919003499211515836461730625000\) | \([2]\) | \(509607936\) | \(4.6099\) | |
| 242550.gw3 | 242550gw5 | \([1, -1, 0, -4826453067, -11331301534659]\) | \(9278380528613437145689/5328033205714065000\) | \(7140074072082662346747890625000\) | \([2]\) | \(509607936\) | \(4.6099\) | |
| 242550.gw2 | 242550gw6 | \([1, -1, 0, -17821565442, -865569702658284]\) | \(467116778179943012100169/28800309694464000000\) | \(38595169470201132096000000000000\) | \([2, 2]\) | \(764411904\) | \(4.8126\) | |
| 242550.gw8 | 242550gw7 | \([1, -1, 0, 13930434558, -3614372094658284]\) | \(223090928422700449019831/4340371122724101696000\) | \(-5816512420257205557428769000000000\) | \([2]\) | \(1528823808\) | \(5.1592\) | |
| 242550.gw1 | 242550gw8 | \([1, -1, 0, -280784573442, -57267190473538284]\) | \(1826870018430810435423307849/7641104625000000000\) | \(10239810997522025390625000000000\) | \([2]\) | \(1528823808\) | \(5.1592\) |
Rank
sage: E.rank()
The elliptic curves in class 242550gw have rank \(0\).
Complex multiplication
The elliptic curves in class 242550gw do not have complex multiplication.Modular form 242550.2.a.gw
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
