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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 69819.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69819.b1 | 69819e2 | \([0, 1, 1, -81227, -8942473]\) | \(-23100424192/14739\) | \(-37816241542251\) | \([]\) | \(308448\) | \(1.5462\) | |
69819.b2 | 69819e1 | \([0, 1, 1, 913, -50818]\) | \(32768/459\) | \(-1177668421731\) | \([]\) | \(102816\) | \(0.99693\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 69819.b have rank \(1\).
Complex multiplication
The elliptic curves in class 69819.b do not have complex multiplication.Modular form 69819.2.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.