Show commands:
SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 67158u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67158.w3 | 67158u1 | \([1, -1, 0, -18936, 1007680]\) | \(1030086793846657/25788672\) | \(18799941888\) | \([2]\) | \(122880\) | \(1.0799\) | \(\Gamma_0(N)\)-optimal |
| 67158.w2 | 67158u2 | \([1, -1, 0, -19656, 927472]\) | \(1152110255377537/162367090704\) | \(118365609123216\) | \([2, 2]\) | \(245760\) | \(1.4265\) | |
| 67158.w4 | 67158u3 | \([1, -1, 0, 32004, 4946620]\) | \(4972803928432703/17424902388348\) | \(-12702753841105692\) | \([2]\) | \(491520\) | \(1.7730\) | |
| 67158.w1 | 67158u4 | \([1, -1, 0, -82836, -8233628]\) | \(86229623764904257/9525651634044\) | \(6944200041218076\) | \([2]\) | \(491520\) | \(1.7730\) |
Rank
sage: E.rank()
The elliptic curves in class 67158u have rank \(0\).
Complex multiplication
The elliptic curves in class 67158u do not have complex multiplication.Modular form 67158.2.a.u
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.
