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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 69300a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69300.f3 | 69300a1 | \([0, 0, 0, -25800, -128875]\) | \(281370820608/161767375\) | \(1091929781250000\) | \([2]\) | \(248832\) | \(1.5752\) | \(\Gamma_0(N)\)-optimal |
| 69300.f4 | 69300a2 | \([0, 0, 0, 102825, -1029250]\) | \(1113258734352/648484375\) | \(-70036312500000000\) | \([2]\) | \(497664\) | \(1.9217\) | |
| 69300.f1 | 69300a3 | \([0, 0, 0, -1495800, -704136375]\) | \(75216478666752/326095\) | \(1604631971250000\) | \([2]\) | \(746496\) | \(2.1245\) | |
| 69300.f2 | 69300a4 | \([0, 0, 0, -1472175, -727454250]\) | \(-4481782160112/310023175\) | \(-24408744614100000000\) | \([2]\) | \(1492992\) | \(2.4711\) |
Rank
sage: E.rank()
The elliptic curves in class 69300a have rank \(0\).
Complex multiplication
The elliptic curves in class 69300a do not have complex multiplication.Modular form 69300.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
