Show commands:
SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 47040.gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.gp1 | 47040dc8 | \([0, 1, 0, -1101465185, 14069962595583]\) | \(4791901410190533590281/41160000\) | \(1269414714408960000\) | \([2]\) | \(10616832\) | \(3.5149\) | |
| 47040.gp2 | 47040dc6 | \([0, 1, 0, -68843105, 219815685375]\) | \(1169975873419524361/108425318400\) | \(3343943017284658790400\) | \([2, 2]\) | \(5308416\) | \(3.1684\) | |
| 47040.gp3 | 47040dc7 | \([0, 1, 0, -63825505, 253221862655]\) | \(-932348627918877961/358766164249920\) | \(-11064699901139704369643520\) | \([2]\) | \(10616832\) | \(3.5149\) | |
| 47040.gp4 | 47040dc5 | \([0, 1, 0, -13665185, 19096955583]\) | \(9150443179640281/184570312500\) | \(5692329216000000000000\) | \([2]\) | \(3538944\) | \(2.9656\) | |
| 47040.gp5 | 47040dc3 | \([0, 1, 0, -4617825, 2901224703]\) | \(353108405631241/86318776320\) | \(2662155607152179281920\) | \([2]\) | \(2654208\) | \(2.8218\) | |
| 47040.gp6 | 47040dc2 | \([0, 1, 0, -1811105, -493097025]\) | \(21302308926361/8930250000\) | \(275417656786944000000\) | \([2, 2]\) | \(1769472\) | \(2.6191\) | |
| 47040.gp7 | 47040dc1 | \([0, 1, 0, -1560225, -750349377]\) | \(13619385906841/6048000\) | \(186526243749888000\) | \([2]\) | \(884736\) | \(2.2725\) | \(\Gamma_0(N)\)-optimal |
| 47040.gp8 | 47040dc4 | \([0, 1, 0, 6028895, -3614985025]\) | \(785793873833639/637994920500\) | \(-19676388236172853248000\) | \([2]\) | \(3538944\) | \(2.9656\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.gp have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.gp do not have complex multiplication.Modular form 47040.2.a.gp
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.
