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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 47040do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.fv5 | 47040do1 | \([0, 1, 0, -3005, -261885]\) | \(-24918016/229635\) | \(-27664719989760\) | \([2]\) | \(98304\) | \(1.2617\) | \(\Gamma_0(N)\)-optimal |
| 47040.fv4 | 47040do2 | \([0, 1, 0, -82385, -9104817]\) | \(32082281296/99225\) | \(191262261657600\) | \([2, 2]\) | \(196608\) | \(1.6083\) | |
| 47040.fv3 | 47040do3 | \([0, 1, 0, -117665, -588225]\) | \(23366901604/13505625\) | \(104131675791360000\) | \([2, 2]\) | \(393216\) | \(1.9549\) | |
| 47040.fv1 | 47040do4 | \([0, 1, 0, -1317185, -582298977]\) | \(32779037733124/315\) | \(2428727132160\) | \([2]\) | \(393216\) | \(1.9549\) | |
| 47040.fv6 | 47040do5 | \([0, 1, 0, 470335, -4233825]\) | \(746185003198/432360075\) | \(-6667204095334809600\) | \([2]\) | \(786432\) | \(2.3014\) | |
| 47040.fv2 | 47040do6 | \([0, 1, 0, -1270145, 548683743]\) | \(14695548366242/57421875\) | \(885473433600000000\) | \([2]\) | \(786432\) | \(2.3014\) |
Rank
sage: E.rank()
The elliptic curves in class 47040do have rank \(0\).
Complex multiplication
The elliptic curves in class 47040do do not have complex multiplication.Modular form 47040.2.a.do
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
