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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 190575.dx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 190575.dx1 | 190575do6 | \([1, -1, 0, -83014470567, -9206145580482284]\) | \(3135316978843283198764801/571725\) | \(11536945696508203125\) | \([2]\) | \(176947200\) | \(4.4521\) | |
| 190575.dx2 | 190575do4 | \([1, -1, 0, -5188404942, -143845020910409]\) | \(765458482133960722801/326869475625\) | \(6595960278336152431640625\) | \([2, 2]\) | \(88473600\) | \(4.1055\) | |
| 190575.dx3 | 190575do5 | \([1, -1, 0, -5162677317, -145342188592034]\) | \(-754127868744065783521/15825714261328125\) | \(-319350047735183888067626953125\) | \([2]\) | \(176947200\) | \(4.4521\) | |
| 190575.dx4 | 190575do3 | \([1, -1, 0, -692740692, 3701012243341]\) | \(1821931919215868881/761147600816295\) | \(15359339783366300714919609375\) | \([2]\) | \(88473600\) | \(4.1055\) | |
| 190575.dx5 | 190575do2 | \([1, -1, 0, -325883817, -2224093144784]\) | \(189674274234120481/3859869269025\) | \(77889023835519240003515625\) | \([2, 2]\) | \(44236800\) | \(3.7590\) | |
| 190575.dx6 | 190575do1 | \([1, -1, 0, 952308, -103907201909]\) | \(4733169839/231139696095\) | \(-4664211154235732516484375\) | \([2]\) | \(22118400\) | \(3.4124\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 190575.dx have rank \(0\).
Complex multiplication
The elliptic curves in class 190575.dx do not have complex multiplication.Modular form 190575.2.a.dx
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
