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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 75690a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75690.k3 | 75690a1 | \([1, -1, 0, -6465, 229501]\) | \(-1860867/320\) | \(-5139273493440\) | \([2]\) | \(193536\) | \(1.1653\) | \(\Gamma_0(N)\)-optimal |
| 75690.k2 | 75690a2 | \([1, -1, 0, -107385, 13571125]\) | \(8527173507/200\) | \(3212045933400\) | \([2]\) | \(387072\) | \(1.5119\) | |
| 75690.k4 | 75690a3 | \([1, -1, 0, 43995, -978175]\) | \(804357/500\) | \(-5853953713621500\) | \([2]\) | \(580608\) | \(1.7146\) | |
| 75690.k1 | 75690a4 | \([1, -1, 0, -183075, -7835689]\) | \(57960603/31250\) | \(365872107101343750\) | \([2]\) | \(1161216\) | \(2.0612\) |
Rank
sage: E.rank()
The elliptic curves in class 75690a have rank \(1\).
Complex multiplication
The elliptic curves in class 75690a do not have complex multiplication.Modular form 75690.2.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.
