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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 90009b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
90009.b2 | 90009b1 | \([1, -1, 1, -482348, -128819546]\) | \(17024594875172176761/13702740137\) | \(9989297559873\) | \([2]\) | \(746496\) | \(1.7996\) | \(\Gamma_0(N)\)-optimal |
90009.b1 | 90009b2 | \([1, -1, 1, -485633, -126973376]\) | \(17374804109361438921/482665506294457\) | \(351863154088659153\) | \([2]\) | \(1492992\) | \(2.1462\) |
Rank
sage: E.rank()
The elliptic curves in class 90009b have rank \(1\).
Complex multiplication
The elliptic curves in class 90009b do not have complex multiplication.Modular form 90009.2.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 Cremona 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 Cremona labels.