Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1690.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1690.f1 | 1690f3 | \([1, 0, 0, -35071, -2429415]\) | \(988345570681/44994560\) | \(217180147159040\) | \([2]\) | \(12096\) | \(1.5129\) | |
| 1690.f2 | 1690f1 | \([1, 0, 0, -5496, 155440]\) | \(3803721481/26000\) | \(125497034000\) | \([2]\) | \(4032\) | \(0.96362\) | \(\Gamma_0(N)\)-optimal |
| 1690.f3 | 1690f2 | \([1, 0, 0, -2116, 345396]\) | \(-217081801/10562500\) | \(-50983170062500\) | \([2]\) | \(8064\) | \(1.3102\) | |
| 1690.f4 | 1690f4 | \([1, 0, 0, 19009, -9232679]\) | \(157376536199/7722894400\) | \(-37276936195969600\) | \([2]\) | \(24192\) | \(1.8595\) |
Rank
sage: E.rank()
The elliptic curves in class 1690.f have rank \(0\).
Complex multiplication
The elliptic curves in class 1690.f do not have complex multiplication.Modular form 1690.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
