Show commands:
SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 269120.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
269120.t1 | 269120t4 | \([0, 0, 0, -359948, -83117712]\) | \(132304644/5\) | \(194911705825280\) | \([2]\) | \(1605632\) | \(1.8280\) | |
269120.t2 | 269120t2 | \([0, 0, 0, -23548, -1170672]\) | \(148176/25\) | \(243639632281600\) | \([2, 2]\) | \(802816\) | \(1.4814\) | |
269120.t3 | 269120t1 | \([0, 0, 0, -6728, 195112]\) | \(55296/5\) | \(3045495403520\) | \([2]\) | \(401408\) | \(1.1349\) | \(\Gamma_0(N)\)-optimal |
269120.t4 | 269120t3 | \([0, 0, 0, 43732, -6633808]\) | \(237276/625\) | \(-24363963228160000\) | \([2]\) | \(1605632\) | \(1.8280\) |
Rank
sage: E.rank()
The elliptic curves in class 269120.t have rank \(1\).
Complex multiplication
The elliptic curves in class 269120.t do not have complex multiplication.Modular form 269120.2.a.t
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.