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SageMath
E = EllipticCurve("ij1")
E.isogeny_class()
Elliptic curves in class 169050ij
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.z4 | 169050ij1 | \([1, 1, 0, 3650, -81500]\) | \(2924207/3312\) | \(-6088335750000\) | \([2]\) | \(393216\) | \(1.1395\) | \(\Gamma_0(N)\)-optimal |
| 169050.z3 | 169050ij2 | \([1, 1, 0, -20850, -792000]\) | \(545338513/171396\) | \(315071375062500\) | \([2, 2]\) | \(786432\) | \(1.4861\) | |
| 169050.z2 | 169050ij3 | \([1, 1, 0, -131100, 17619750]\) | \(135559106353/5037138\) | \(9259597633781250\) | \([2]\) | \(1572864\) | \(1.8327\) | |
| 169050.z1 | 169050ij4 | \([1, 1, 0, -302600, -64185750]\) | \(1666957239793/301806\) | \(554799595218750\) | \([2]\) | \(1572864\) | \(1.8327\) |
Rank
sage: E.rank()
The elliptic curves in class 169050ij have rank \(1\).
Complex multiplication
The elliptic curves in class 169050ij do not have complex multiplication.Modular form 169050.2.a.ij
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.
