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SageMath
E = EllipticCurve("gf1")
E.isogeny_class()
Elliptic curves in class 187200gf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.ph4 | 187200gf1 | \([0, 0, 0, -4154700, 5384954000]\) | \(-2656166199049/2658140160\) | \(-7937163987517440000000\) | \([2]\) | \(11796480\) | \(2.8979\) | \(\Gamma_0(N)\)-optimal |
| 187200.ph3 | 187200gf2 | \([0, 0, 0, -77882700, 264465146000]\) | \(17496824387403529/6580454400\) | \(19649131551129600000000\) | \([2, 2]\) | \(23592960\) | \(3.2444\) | |
| 187200.ph1 | 187200gf3 | \([0, 0, 0, -1246010700, 16928979194000]\) | \(71647584155243142409/10140000\) | \(30277877760000000000\) | \([2]\) | \(47185920\) | \(3.5910\) | |
| 187200.ph2 | 187200gf4 | \([0, 0, 0, -89402700, 181083386000]\) | \(26465989780414729/10571870144160\) | \(31567435100539453440000000\) | \([2]\) | \(47185920\) | \(3.5910\) |
Rank
sage: E.rank()
The elliptic curves in class 187200gf have rank \(1\).
Complex multiplication
The elliptic curves in class 187200gf do not have complex multiplication.Modular form 187200.2.a.gf
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.
