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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 16731.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16731.k1 | 16731k3 | \([1, -1, 0, -222858, -40398539]\) | \(347873904937/395307\) | \(1390984039929627\) | \([2]\) | \(110592\) | \(1.8191\) | |
| 16731.k2 | 16731k2 | \([1, -1, 0, -17523, -276080]\) | \(169112377/88209\) | \(310384868414049\) | \([2, 2]\) | \(55296\) | \(1.4726\) | |
| 16731.k3 | 16731k1 | \([1, -1, 0, -9918, 379471]\) | \(30664297/297\) | \(1045066897017\) | \([2]\) | \(27648\) | \(1.1260\) | \(\Gamma_0(N)\)-optimal |
| 16731.k4 | 16731k4 | \([1, -1, 0, 66132, -2200145]\) | \(9090072503/5845851\) | \(-20570051733985611\) | \([2]\) | \(110592\) | \(1.8191\) |
Rank
sage: E.rank()
The elliptic curves in class 16731.k have rank \(1\).
Complex multiplication
The elliptic curves in class 16731.k do not have complex multiplication.Modular form 16731.2.a.k
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.
