Show commands:
SageMath
E = EllipticCurve("hx1")
E.isogeny_class()
Elliptic curves in class 127050.hx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.hx1 | 127050hg4 | \([1, 0, 0, -99797838, -383737186458]\) | \(3971101377248209009/56495958750\) | \(1563844330923574218750\) | \([2]\) | \(17694720\) | \(3.2065\) | |
| 127050.hx2 | 127050hg2 | \([1, 0, 0, -6416088, -5634480708]\) | \(1055257664218129/115307784900\) | \(3191793355081701562500\) | \([2, 2]\) | \(8847360\) | \(2.8599\) | |
| 127050.hx3 | 127050hg1 | \([1, 0, 0, -1515588, 623457792]\) | \(13908844989649/1980372240\) | \(54817972279166250000\) | \([4]\) | \(4423680\) | \(2.5133\) | \(\Gamma_0(N)\)-optimal |
| 127050.hx4 | 127050hg3 | \([1, 0, 0, 8557662, -28020236958]\) | \(2503876820718671/13702874328990\) | \(-379304339830310209218750\) | \([2]\) | \(17694720\) | \(3.2065\) |
Rank
sage: E.rank()
The elliptic curves in class 127050.hx have rank \(0\).
Complex multiplication
The elliptic curves in class 127050.hx do not have complex multiplication.Modular form 127050.2.a.hx
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.
