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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 19602d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19602.i3 | 19602d1 | \([1, -1, 0, -567, -5307]\) | \(-140625/8\) | \(-1147971528\) | \([]\) | \(8100\) | \(0.49536\) | \(\Gamma_0(N)\)-optimal |
| 19602.i4 | 19602d2 | \([1, -1, 0, 3063, -10873]\) | \(3375/2\) | \(-1882960298802\) | \([]\) | \(24300\) | \(1.0447\) | |
| 19602.i2 | 19602d3 | \([1, -1, 0, -11457, 961725]\) | \(-1159088625/2097152\) | \(-300933848236032\) | \([]\) | \(56700\) | \(1.4683\) | |
| 19602.i1 | 19602d4 | \([1, -1, 0, -1173057, 489313853]\) | \(-189613868625/128\) | \(-120509459123328\) | \([]\) | \(170100\) | \(2.0176\) |
Rank
sage: E.rank()
The elliptic curves in class 19602d have rank \(0\).
Complex multiplication
The elliptic curves in class 19602d do not have complex multiplication.Modular form 19602.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
