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SageMath
E = EllipticCurve("gw1")
E.isogeny_class()
Elliptic curves in class 88200gw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 88200.hd3 | 88200gw1 | \([0, 0, 0, -786450, -257292875]\) | \(2508888064/118125\) | \(2532780760781250000\) | \([2]\) | \(1769472\) | \(2.2921\) | \(\Gamma_0(N)\)-optimal |
| 88200.hd2 | 88200gw2 | \([0, 0, 0, -2164575, 890685250]\) | \(3269383504/893025\) | \(306365160824100000000\) | \([2, 2]\) | \(3538944\) | \(2.6387\) | |
| 88200.hd4 | 88200gw3 | \([0, 0, 0, 5552925, 5806732750]\) | \(13799183324/18600435\) | \(-25524594541802160000000\) | \([2]\) | \(7077888\) | \(2.9853\) | |
| 88200.hd1 | 88200gw4 | \([0, 0, 0, -31932075, 69445237750]\) | \(2624033547076/324135\) | \(444796826085360000000\) | \([2]\) | \(7077888\) | \(2.9853\) |
Rank
sage: E.rank()
The elliptic curves in class 88200gw have rank \(0\).
Complex multiplication
The elliptic curves in class 88200gw do not have complex multiplication.Modular form 88200.2.a.gw
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.
