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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 20160.et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.et1 | 20160cg7 | \([0, 0, 0, -202309932, 1107576977456]\) | \(4791901410190533590281/41160000\) | \(7865798492160000\) | \([2]\) | \(1769472\) | \(3.0913\) | |
| 20160.et2 | 20160cg6 | \([0, 0, 0, -12644652, 17305081904]\) | \(1169975873419524361/108425318400\) | \(20720401019987558400\) | \([2, 2]\) | \(884736\) | \(2.7447\) | |
| 20160.et3 | 20160cg8 | \([0, 0, 0, -11723052, 19934591024]\) | \(-932348627918877961/358766164249920\) | \(-68561281676264519761920\) | \([2]\) | \(1769472\) | \(3.0913\) | |
| 20160.et4 | 20160cg4 | \([0, 0, 0, -2509932, 1503617456]\) | \(9150443179640281/184570312500\) | \(35271936000000000000\) | \([2]\) | \(589824\) | \(2.5420\) | |
| 20160.et5 | 20160cg3 | \([0, 0, 0, -848172, 228497456]\) | \(353108405631241/86318776320\) | \(16495775039430328320\) | \([2]\) | \(442368\) | \(2.3981\) | |
| 20160.et6 | 20160cg2 | \([0, 0, 0, -332652, -38767696]\) | \(21302308926361/8930250000\) | \(1706597351424000000\) | \([2, 2]\) | \(294912\) | \(2.1954\) | |
| 20160.et7 | 20160cg1 | \([0, 0, 0, -286572, -59024464]\) | \(13619385906841/6048000\) | \(1155790798848000\) | \([2]\) | \(147456\) | \(1.8488\) | \(\Gamma_0(N)\)-optimal |
| 20160.et8 | 20160cg5 | \([0, 0, 0, 1107348, -284719696]\) | \(785793873833639/637994920500\) | \(-121922727980433408000\) | \([2]\) | \(589824\) | \(2.5420\) |
Rank
sage: E.rank()
The elliptic curves in class 20160.et have rank \(0\).
Complex multiplication
The elliptic curves in class 20160.et do not have complex multiplication.Modular form 20160.2.a.et
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
