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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 7245s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7245.s4 | 7245s1 | \([1, -1, 0, -2079, -9072]\) | \(1363569097969/734582625\) | \(535510733625\) | \([2]\) | \(8448\) | \(0.94164\) | \(\Gamma_0(N)\)-optimal |
7245.s2 | 7245s2 | \([1, -1, 0, -25884, -1594485]\) | \(2630872462131649/3645140625\) | \(2657307515625\) | \([2, 2]\) | \(16896\) | \(1.2882\) | |
7245.s1 | 7245s3 | \([1, -1, 0, -414009, -102429360]\) | \(10765299591712341649/20708625\) | \(15096587625\) | \([2]\) | \(33792\) | \(1.6348\) | |
7245.s3 | 7245s4 | \([1, -1, 0, -18639, -2511702]\) | \(-982374577874929/3183837890625\) | \(-2321017822265625\) | \([2]\) | \(33792\) | \(1.6348\) |
Rank
sage: E.rank()
The elliptic curves in class 7245s have rank \(0\).
Complex multiplication
The elliptic curves in class 7245s do not have complex multiplication.Modular form 7245.2.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.