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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 53130.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53130.cs1 | 53130cr8 | \([1, 0, 0, -2418345370, -45774911264488]\) | \(1564151023065118902972769846162081/22955387647500\) | \(22955387647500\) | \([2]\) | \(15925248\) | \(3.5397\) | |
| 53130.cs2 | 53130cr7 | \([1, 0, 0, -151545890, -711273429360]\) | \(384907346279514350400752451361/4201994679045883946979060\) | \(4201994679045883946979060\) | \([2]\) | \(15925248\) | \(3.5397\) | |
| 53130.cs3 | 53130cr6 | \([1, 0, 0, -151146590, -715242391500]\) | \(381872841673941838282574504161/46261712772301395600\) | \(46261712772301395600\) | \([2, 2]\) | \(7962624\) | \(3.1932\) | |
| 53130.cs4 | 53130cr5 | \([1, 0, 0, -29857870, -62785836988]\) | \(2943744963623989512239962081/720592477171875000000\) | \(720592477171875000000\) | \([6]\) | \(5308416\) | \(2.9904\) | |
| 53130.cs5 | 53130cr4 | \([1, 0, 0, -13968590, 19556060100]\) | \(301426495993762500163592161/9203102532237337416000\) | \(9203102532237337416000\) | \([6]\) | \(5308416\) | \(2.9904\) | |
| 53130.cs6 | 53130cr3 | \([1, 0, 0, -9421710, -11238222588]\) | \(-92493861830012244531817441/1026419736641091260160\) | \(-1026419736641091260160\) | \([2]\) | \(3981312\) | \(2.8466\) | |
| 53130.cs7 | 53130cr2 | \([1, 0, 0, -2088590, -732603900]\) | \(1007588745830352584072161/351347432901696000000\) | \(351347432901696000000\) | \([2, 6]\) | \(2654208\) | \(2.6439\) | |
| 53130.cs8 | 53130cr1 | \([1, 0, 0, 389490, -79877628]\) | \(6534503243692176483359/6540129831223296000\) | \(-6540129831223296000\) | \([6]\) | \(1327104\) | \(2.2973\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53130.cs have rank \(1\).
Complex multiplication
The elliptic curves in class 53130.cs do not have complex multiplication.Modular form 53130.2.a.cs
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
