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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 528.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
528.j1 | 528j3 | \([0, 1, 0, -5632, 160820]\) | \(4824238966273/66\) | \(270336\) | \([2]\) | \(384\) | \(0.59819\) | |
528.j2 | 528j2 | \([0, 1, 0, -352, 2420]\) | \(1180932193/4356\) | \(17842176\) | \([2, 2]\) | \(192\) | \(0.25162\) | |
528.j3 | 528j4 | \([0, 1, 0, -192, 4788]\) | \(-192100033/2371842\) | \(-9715064832\) | \([4]\) | \(384\) | \(0.59819\) | |
528.j4 | 528j1 | \([0, 1, 0, -32, -12]\) | \(912673/528\) | \(2162688\) | \([2]\) | \(96\) | \(-0.094954\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 528.j have rank \(0\).
Complex multiplication
The elliptic curves in class 528.j do not have complex multiplication.Modular form 528.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.