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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 12870.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12870.s1 | 12870z2 | \([1, -1, 0, -263934, 52256488]\) | \(2789222297765780449/677605500\) | \(493974409500\) | \([2]\) | \(73728\) | \(1.6211\) | |
12870.s2 | 12870z1 | \([1, -1, 0, -16434, 825988]\) | \(-673350049820449/10617750000\) | \(-7740339750000\) | \([2]\) | \(36864\) | \(1.2745\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.s have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.s do not have complex multiplication.Modular form 12870.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.