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SageMath
sage: E = EllipticCurve("h1")
sage: E.isogeny_class()
Elliptic curves in class 1008h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
1008.d4 | 1008h1 | [0, 0, 0, 9, 54] | [2] | 128 | \(\Gamma_0(N)\)-optimal |
1008.d3 | 1008h2 | [0, 0, 0, -171, 810] | [2, 2] | 256 | |
1008.d2 | 1008h3 | [0, 0, 0, -531, -3726] | [2] | 512 | |
1008.d1 | 1008h4 | [0, 0, 0, -2691, 53730] | [2] | 512 |
Rank
sage: E.rank()
The elliptic curves in class 1008h have rank \(1\).
Complex multiplication
The elliptic curves in class 1008h do not have complex multiplication.Modular form 1008.2.a.h
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.