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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 850.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
850.b1 | 850a2 | \([1, 1, 0, -166025, -26946875]\) | \(-32391289681150609/1228250000000\) | \(-19191406250000000\) | \([]\) | \(6048\) | \(1.8945\) | |
850.b2 | 850a1 | \([1, 1, 0, 9975, -114875]\) | \(7023836099951/4456448000\) | \(-69632000000000\) | \([]\) | \(2016\) | \(1.3452\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 850.b have rank \(0\).
Complex multiplication
The elliptic curves in class 850.b do not have complex multiplication.Modular form 850.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.