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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 154560.dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 154560.dr1 | 154560gi7 | \([0, -1, 0, -16249537025, -748042877349375]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(32864587875000000000000000000000\) | \([4]\) | \(382205952\) | \(4.7955\) | |
| 154560.dr2 | 154560gi6 | \([0, -1, 0, -15969209345, -776726734418943]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(2406623000334336000000000000\) | \([2, 2]\) | \(191102976\) | \(4.4489\) | |
| 154560.dr3 | 154560gi3 | \([0, -1, 0, -15969188865, -776728826315775]\) | \(1718036403880129446396978632449/49057344000000\) | \(12860088385536000000\) | \([2]\) | \(95551488\) | \(4.1023\) | |
| 154560.dr4 | 154560gi8 | \([0, -1, 0, -15689209345, -805276710418943]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-32980105831690970155155456000000\) | \([2]\) | \(382205952\) | \(4.7955\) | |
| 154560.dr5 | 154560gi4 | \([0, -1, 0, -3028286465, 63923231783937]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(12167511408596326809600000000\) | \([4]\) | \(127401984\) | \(4.2461\) | |
| 154560.dr6 | 154560gi2 | \([0, -1, 0, -281181185, -71684555775]\) | \(9378698233516887309850369/5418996968417034240000\) | \(1420557541288715023810560000\) | \([2, 2]\) | \(63700992\) | \(3.8996\) | |
| 154560.dr7 | 154560gi1 | \([0, -1, 0, -197295105, -1063771669503]\) | \(3239908336204082689644289/9880281924658790400\) | \(2590056624857753950617600\) | \([2]\) | \(31850496\) | \(3.5530\) | \(\Gamma_0(N)\)-optimal |
| 154560.dr8 | 154560gi5 | \([0, -1, 0, 1123746815, -574367794175]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-90963145325872008134236569600\) | \([2]\) | \(127401984\) | \(4.2461\) |
Rank
sage: E.rank()
The elliptic curves in class 154560.dr have rank \(1\).
Complex multiplication
The elliptic curves in class 154560.dr do not have complex multiplication.Modular form 154560.2.a.dr
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
