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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 528.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
528.i1 | 528d3 | \([0, 1, 0, -472, -4108]\) | \(5690357426/891\) | \(1824768\) | \([2]\) | \(128\) | \(0.21241\) | |
528.i2 | 528d2 | \([0, 1, 0, -32, -60]\) | \(3650692/1089\) | \(1115136\) | \([2, 2]\) | \(64\) | \(-0.13416\) | |
528.i3 | 528d1 | \([0, 1, 0, -12, 12]\) | \(810448/33\) | \(8448\) | \([2]\) | \(32\) | \(-0.48073\) | \(\Gamma_0(N)\)-optimal |
528.i4 | 528d4 | \([0, 1, 0, 88, -300]\) | \(36382894/43923\) | \(-89954304\) | \([2]\) | \(128\) | \(0.21241\) |
Rank
sage: E.rank()
The elliptic curves in class 528.i have rank \(0\).
Complex multiplication
The elliptic curves in class 528.i do not have complex multiplication.Modular form 528.2.a.i
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.