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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 3648r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3648.x4 | 3648r1 | \([0, 1, 0, -22529, -1176993]\) | \(4824238966273/537919488\) | \(141012366262272\) | \([2]\) | \(11520\) | \(1.4480\) | \(\Gamma_0(N)\)-optimal |
| 3648.x2 | 3648r2 | \([0, 1, 0, -350209, -79885729]\) | \(18120364883707393/269485056\) | \(70643890520064\) | \([2, 2]\) | \(23040\) | \(1.7946\) | |
| 3648.x1 | 3648r3 | \([0, 1, 0, -5603329, -5107121569]\) | \(74220219816682217473/16416\) | \(4303355904\) | \([2]\) | \(46080\) | \(2.1412\) | |
| 3648.x3 | 3648r4 | \([0, 1, 0, -339969, -84766113]\) | \(-16576888679672833/2216253521952\) | \(-580977563258585088\) | \([4]\) | \(46080\) | \(2.1412\) |
Rank
sage: E.rank()
The elliptic curves in class 3648r have rank \(1\).
Complex multiplication
The elliptic curves in class 3648r do not have complex multiplication.Modular form 3648.2.a.r
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.
