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SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 388416fe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388416.fe4 | 388416fe1 | \([0, 1, 0, 18111, -253475649]\) | \(103823/4386816\) | \(-27757661097899851776\) | \([2]\) | \(10616832\) | \(2.4097\) | \(\Gamma_0(N)\)-optimal |
| 388416.fe3 | 388416fe2 | \([0, 1, 0, -5900609, -5420518209]\) | \(3590714269297/73410624\) | \(464507109935167832064\) | \([2, 2]\) | \(21233664\) | \(2.7563\) | |
| 388416.fe2 | 388416fe3 | \([0, 1, 0, -12559169, 9051196095]\) | \(34623662831857/14438442312\) | \(91359516441554328748032\) | \([2]\) | \(42467328\) | \(3.1029\) | |
| 388416.fe1 | 388416fe4 | \([0, 1, 0, -93941569, -350488256833]\) | \(14489843500598257/6246072\) | \(39522138555408187392\) | \([2]\) | \(42467328\) | \(3.1029\) |
Rank
sage: E.rank()
The elliptic curves in class 388416fe have rank \(1\).
Complex multiplication
The elliptic curves in class 388416fe do not have complex multiplication.Modular form 388416.2.a.fe
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.
