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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 101640.ct
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 101640.ct1 | 101640bm4 | \([0, 1, 0, -813160, -282507040]\) | \(32779037733124/315\) | \(571434716160\) | \([2]\) | \(655360\) | \(1.8343\) | |
| 101640.ct2 | 101640bm6 | \([0, 1, 0, -784120, 266087600]\) | \(14695548366242/57421875\) | \(208335573600000000\) | \([2]\) | \(1310720\) | \(2.1809\) | |
| 101640.ct3 | 101640bm3 | \([0, 1, 0, -72640, -290512]\) | \(23366901604/13505625\) | \(24500263455360000\) | \([2, 2]\) | \(655360\) | \(1.8343\) | |
| 101640.ct4 | 101640bm2 | \([0, 1, 0, -50860, -4420000]\) | \(32082281296/99225\) | \(45000483897600\) | \([2, 2]\) | \(327680\) | \(1.4877\) | |
| 101640.ct5 | 101640bm1 | \([0, 1, 0, -1855, -127162]\) | \(-24918016/229635\) | \(-6508998563760\) | \([2]\) | \(163840\) | \(1.1411\) | \(\Gamma_0(N)\)-optimal |
| 101640.ct6 | 101640bm5 | \([0, 1, 0, 290360, -2032912]\) | \(746185003198/432360075\) | \(-1568670201501849600\) | \([2]\) | \(1310720\) | \(2.1809\) |
Rank
sage: E.rank()
The elliptic curves in class 101640.ct have rank \(1\).
Complex multiplication
The elliptic curves in class 101640.ct do not have complex multiplication.Modular form 101640.2.a.ct
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
