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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 528.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
528.a1 | 528f3 | \([0, -1, 0, -161040, 24927936]\) | \(112763292123580561/1932612\) | \(7915978752\) | \([2]\) | \(2400\) | \(1.4410\) | |
528.a2 | 528f4 | \([0, -1, 0, -160880, 24979776]\) | \(-112427521449300721/466873642818\) | \(-1912314440982528\) | \([2]\) | \(4800\) | \(1.7876\) | |
528.a3 | 528f1 | \([0, -1, 0, -720, -5184]\) | \(10091699281/2737152\) | \(11211374592\) | \([2]\) | \(480\) | \(0.63628\) | \(\Gamma_0(N)\)-optimal |
528.a4 | 528f2 | \([0, -1, 0, 1840, -35904]\) | \(168105213359/228637728\) | \(-936500133888\) | \([2]\) | \(960\) | \(0.98285\) |
Rank
sage: E.rank()
The elliptic curves in class 528.a have rank \(0\).
Complex multiplication
The elliptic curves in class 528.a do not have complex multiplication.Modular form 528.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.