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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 170352.fn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 170352.fn1 | 170352cg3 | \([0, 0, 0, -10148619, 12443706938]\) | \(8020417344913/187278\) | \(2699195572521197568\) | \([4]\) | \(6193152\) | \(2.6489\) | |
| 170352.fn2 | 170352cg2 | \([0, 0, 0, -657579, 179385050]\) | \(2181825073/298116\) | \(4296678666462314496\) | \([2, 2]\) | \(3096576\) | \(2.3023\) | |
| 170352.fn3 | 170352cg1 | \([0, 0, 0, -170859, -24355942]\) | \(38272753/4368\) | \(62954998776004608\) | \([2]\) | \(1548288\) | \(1.9558\) | \(\Gamma_0(N)\)-optimal |
| 170352.fn4 | 170352cg4 | \([0, 0, 0, 1045941, 954486650]\) | \(8780064047/32388174\) | \(-466803446549227167744\) | \([2]\) | \(6193152\) | \(2.6489\) |
Rank
sage: E.rank()
The elliptic curves in class 170352.fn have rank \(0\).
Complex multiplication
The elliptic curves in class 170352.fn do not have complex multiplication.Modular form 170352.2.a.fn
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 LMFDB 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 LMFDB labels.
