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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 129360.ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.ew1 | 129360go8 | \([0, 1, 0, -201039176, 1029330652020]\) | \(1864737106103260904761/129177711985836360\) | \(62249486898879127214653440\) | \([2]\) | \(31850496\) | \(3.6974\) | |
| 129360.ew2 | 129360go5 | \([0, 1, 0, -197569976, 1068815581140]\) | \(1769857772964702379561/691787250\) | \(333365568205824000\) | \([2]\) | \(10616832\) | \(3.1481\) | |
| 129360.ew3 | 129360go6 | \([0, 1, 0, -39691976, -76930290060]\) | \(14351050585434661561/3001282273281600\) | \(1446288827061481301606400\) | \([2, 2]\) | \(15925248\) | \(3.3508\) | |
| 129360.ew4 | 129360go3 | \([0, 1, 0, -37434056, -88162990476]\) | \(12038605770121350841/757333463040\) | \(364951652733718364160\) | \([2]\) | \(7962624\) | \(3.0042\) | |
| 129360.ew5 | 129360go2 | \([0, 1, 0, -12349976, 16691893140]\) | \(432288716775559561/270140062500\) | \(130177876840704000000\) | \([2, 2]\) | \(5308416\) | \(2.8015\) | |
| 129360.ew6 | 129360go4 | \([0, 1, 0, -10021496, 23180901204]\) | \(-230979395175477481/348191894531250\) | \(-167790297906000000000000\) | \([2]\) | \(10616832\) | \(3.1481\) | |
| 129360.ew7 | 129360go1 | \([0, 1, 0, -919256, 153927444]\) | \(178272935636041/81841914000\) | \(39438824817401856000\) | \([2]\) | \(2654208\) | \(2.4549\) | \(\Gamma_0(N)\)-optimal |
| 129360.ew8 | 129360go7 | \([0, 1, 0, 85528504, -464262278796]\) | \(143584693754978072519/276341298967965000\) | \(-133166192567427540111360000\) | \([2]\) | \(31850496\) | \(3.6974\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.ew have rank \(1\).
Complex multiplication
The elliptic curves in class 129360.ew do not have complex multiplication.Modular form 129360.2.a.ew
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
