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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4290a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4290.d4 | 4290a1 | \([1, 1, 0, 22, -2172]\) | \(1095912791/2055596400\) | \(-2055596400\) | \([2]\) | \(3072\) | \(0.46595\) | \(\Gamma_0(N)\)-optimal |
| 4290.d3 | 4290a2 | \([1, 1, 0, -2398, -45248]\) | \(1525998818291689/37268302500\) | \(37268302500\) | \([2, 2]\) | \(6144\) | \(0.81252\) | |
| 4290.d1 | 4290a3 | \([1, 1, 0, -38148, -2883798]\) | \(6139836723518159689/3799803150\) | \(3799803150\) | \([2]\) | \(12288\) | \(1.1591\) | |
| 4290.d2 | 4290a4 | \([1, 1, 0, -5368, 84838]\) | \(17111482619973769/6627044531250\) | \(6627044531250\) | \([2]\) | \(12288\) | \(1.1591\) |
Rank
sage: E.rank()
The elliptic curves in class 4290a have rank \(0\).
Complex multiplication
The elliptic curves in class 4290a do not have complex multiplication.Modular form 4290.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
