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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 21021b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21021.e5 | 21021b1 | \([1, 1, 1, -1177, -22786]\) | \(-1532808577/938223\) | \(-110380997727\) | \([2]\) | \(24576\) | \(0.82090\) | \(\Gamma_0(N)\)-optimal |
| 21021.e4 | 21021b2 | \([1, 1, 1, -21022, -1181734]\) | \(8732907467857/1656369\) | \(194870156481\) | \([2, 2]\) | \(49152\) | \(1.1675\) | |
| 21021.e3 | 21021b3 | \([1, 1, 1, -23227, -921544]\) | \(11779205551777/3763454409\) | \(442766647764441\) | \([2, 2]\) | \(98304\) | \(1.5141\) | |
| 21021.e1 | 21021b4 | \([1, 1, 1, -336337, -75217696]\) | \(35765103905346817/1287\) | \(151414263\) | \([2]\) | \(98304\) | \(1.5141\) | |
| 21021.e6 | 21021b5 | \([1, 1, 1, 65708, -6186496]\) | \(266679605718863/296110251723\) | \(-34837075004959227\) | \([2]\) | \(196608\) | \(1.8606\) | |
| 21021.e2 | 21021b6 | \([1, 1, 1, -147442, 21039668]\) | \(3013001140430737/108679952667\) | \(12786087751319883\) | \([2]\) | \(196608\) | \(1.8606\) |
Rank
sage: E.rank()
The elliptic curves in class 21021b have rank \(0\).
Complex multiplication
The elliptic curves in class 21021b do not have complex multiplication.Modular form 21021.2.a.b
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.
