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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 90.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
90.a1 | 90a4 | \([1, -1, 0, -1149, -14707]\) | \(8527173507/200\) | \(3936600\) | \([2]\) | \(48\) | \(0.37752\) | |
90.a2 | 90a3 | \([1, -1, 0, -69, -235]\) | \(-1860867/320\) | \(-6298560\) | \([2]\) | \(24\) | \(0.030944\) | |
90.a3 | 90a2 | \([1, -1, 0, -24, 18]\) | \(57960603/31250\) | \(843750\) | \([6]\) | \(16\) | \(-0.17179\) | |
90.a4 | 90a1 | \([1, -1, 0, 6, 0]\) | \(804357/500\) | \(-13500\) | \([6]\) | \(8\) | \(-0.51836\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 90.a have rank \(0\).
Complex multiplication
The elliptic curves in class 90.a do not have complex multiplication.Modular form 90.2.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 LMFDB 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 LMFDB labels.