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SageMath
sage: E = EllipticCurve("4032.r1")
sage: E.isogeny_class()
Elliptic curves in class 4032z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
4032.r5 | 4032z1 | [0, 0, 0, -300, 4048] | [2] | 1536 | \(\Gamma_0(N)\)-optimal |
4032.r4 | 4032z2 | [0, 0, 0, -6060, 181456] | [2] | 3072 | |
4032.r6 | 4032z3 | [0, 0, 0, 2580, -84656] | [2] | 4608 | |
4032.r3 | 4032z4 | [0, 0, 0, -20460, -923312] | [2] | 9216 | |
4032.r2 | 4032z5 | [0, 0, 0, -98220, -11882288] | [2] | 13824 | |
4032.r1 | 4032z6 | [0, 0, 0, -1572780, -759189296] | [2] | 27648 |
Rank
sage: E.rank()
The elliptic curves in class 4032z have rank \(1\).
Modular form 4032.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.