Show commands:
SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 8112.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8112.j1 | 8112w4 | \([0, -1, 0, -191974592, -1023732753408]\) | \(18013780041269221/9216\) | \(400306447243542528\) | \([2]\) | \(748800\) | \(3.1456\) | |
| 8112.j2 | 8112w3 | \([0, -1, 0, -11996352, -15998592000]\) | \(-4395631034341/3145728\) | \(-136637933992462516224\) | \([2]\) | \(374400\) | \(2.7991\) | |
| 8112.j3 | 8112w2 | \([0, -1, 0, -571952, 63200448]\) | \(476379541/236196\) | \(10259416407675322368\) | \([2]\) | \(149760\) | \(2.3409\) | |
| 8112.j4 | 8112w1 | \([0, -1, 0, 131088, 7519680]\) | \(5735339/3888\) | \(-168879282430869504\) | \([2]\) | \(74880\) | \(1.9943\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8112.j have rank \(1\).
Complex multiplication
The elliptic curves in class 8112.j do not have complex multiplication.Modular form 8112.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
