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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 28749a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28749.a3 | 28749a1 | \([1, 1, 1, -22617, 1297998]\) | \(498677257/777\) | \(1993569419793\) | \([4]\) | \(65664\) | \(1.2588\) | \(\Gamma_0(N)\)-optimal |
| 28749.a2 | 28749a2 | \([1, 1, 1, -29462, 438266]\) | \(1102302937/603729\) | \(1549003439179161\) | \([2, 2]\) | \(131328\) | \(1.6054\) | |
| 28749.a4 | 28749a3 | \([1, 1, 1, 114283, 3600656]\) | \(64336588343/39357381\) | \(-100980271820774829\) | \([2]\) | \(262656\) | \(1.9520\) | |
| 28749.a1 | 28749a4 | \([1, 1, 1, -282727, -57610072]\) | \(974126411497/7195797\) | \(18462446396702973\) | \([2]\) | \(262656\) | \(1.9520\) |
Rank
sage: E.rank()
The elliptic curves in class 28749a have rank \(1\).
Complex multiplication
The elliptic curves in class 28749a do not have complex multiplication.Modular form 28749.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
