Show commands:
SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 185130dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 185130.cl4 | 185130dl1 | \([1, -1, 0, -31059, 7907125]\) | \(-2565726409/19388160\) | \(-25039187617847040\) | \([2]\) | \(1966080\) | \(1.8296\) | \(\Gamma_0(N)\)-optimal |
| 185130.cl3 | 185130dl2 | \([1, -1, 0, -815139, 282805573]\) | \(46380496070089/125888400\) | \(162580836268659600\) | \([2, 2]\) | \(3932160\) | \(2.1762\) | |
| 185130.cl1 | 185130dl3 | \([1, -1, 0, -13033719, 18114601225]\) | \(189602977175292169/1402500\) | \(1811283826522500\) | \([2]\) | \(7864320\) | \(2.5227\) | |
| 185130.cl2 | 185130dl4 | \([1, -1, 0, -1141839, 35101633]\) | \(127483771761289/73369857660\) | \(94754821057979292540\) | \([2]\) | \(7864320\) | \(2.5227\) |
Rank
sage: E.rank()
The elliptic curves in class 185130dl have rank \(0\).
Complex multiplication
The elliptic curves in class 185130dl do not have complex multiplication.Modular form 185130.2.a.dl
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.
