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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 3570.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3570.s1 | 3570r4 | \([1, 1, 1, -11206, 451919]\) | \(155624507032726369/175394100\) | \(175394100\) | \([2]\) | \(4096\) | \(0.86718\) | |
3570.s2 | 3570r3 | \([1, 1, 1, -1726, -18577]\) | \(568671957006049/191329687500\) | \(191329687500\) | \([2]\) | \(4096\) | \(0.86718\) | |
3570.s3 | 3570r2 | \([1, 1, 1, -706, 6719]\) | \(38920307374369/1274490000\) | \(1274490000\) | \([2, 2]\) | \(2048\) | \(0.52061\) | |
3570.s4 | 3570r1 | \([1, 1, 1, 14, 383]\) | \(302111711/61689600\) | \(-61689600\) | \([2]\) | \(1024\) | \(0.17403\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3570.s have rank \(1\).
Complex multiplication
The elliptic curves in class 3570.s do not have complex multiplication.Modular form 3570.2.a.s
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.