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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 8712.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8712.s1 | 8712a2 | \([0, 0, 0, -171699, 27378670]\) | \(4293378\) | \(130385075521536\) | \([2]\) | \(33792\) | \(1.6992\) | |
8712.s2 | 8712a1 | \([0, 0, 0, -11979, 322102]\) | \(2916\) | \(65192537760768\) | \([2]\) | \(16896\) | \(1.3526\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8712.s have rank \(1\).
Complex multiplication
The elliptic curves in class 8712.s do not have complex multiplication.Modular form 8712.2.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.