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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 289a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 289.a4 | 289a1 | \([1, -1, 1, -199, 510]\) | \(35937/17\) | \(410338673\) | \([4]\) | \(72\) | \(0.34682\) | \(\Gamma_0(N)\)-optimal |
| 289.a2 | 289a2 | \([1, -1, 1, -1644, -24922]\) | \(20346417/289\) | \(6975757441\) | \([2, 2]\) | \(144\) | \(0.69340\) | |
| 289.a1 | 289a3 | \([1, -1, 1, -26209, -1626560]\) | \(82483294977/17\) | \(410338673\) | \([2]\) | \(288\) | \(1.0400\) | |
| 289.a3 | 289a4 | \([1, -1, 1, -199, -68272]\) | \(-35937/83521\) | \(-2015993900449\) | \([2]\) | \(288\) | \(1.0400\) |
Rank
sage: E.rank()
The elliptic curves in class 289a have rank \(1\).
Complex multiplication
The elliptic curves in class 289a do not have complex multiplication.Modular form 289.2.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.
