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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 3570s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3570.r4 | 3570s1 | \([1, 1, 1, 2269, -6847]\) | \(1291859362462031/773834342400\) | \(-773834342400\) | \([2]\) | \(6144\) | \(0.97051\) | \(\Gamma_0(N)\)-optimal |
3570.r3 | 3570s2 | \([1, 1, 1, -9251, -66751]\) | \(87557366190249649/48960807840000\) | \(48960807840000\) | \([2, 2]\) | \(12288\) | \(1.3171\) | |
3570.r1 | 3570s3 | \([1, 1, 1, -111251, -14305951]\) | \(152277495831664137649/282362258900400\) | \(282362258900400\) | \([2]\) | \(24576\) | \(1.6637\) | |
3570.r2 | 3570s4 | \([1, 1, 1, -91571, 10568993]\) | \(84917632843343402929/537144431250000\) | \(537144431250000\) | \([2]\) | \(24576\) | \(1.6637\) |
Rank
sage: E.rank()
The elliptic curves in class 3570s have rank \(1\).
Complex multiplication
The elliptic curves in class 3570s 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 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.