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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 105710.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
105710.b1 | 105710a1 | \([1, 1, 0, -85068, -9676592]\) | \(-76711450249/851840\) | \(-756011135623040\) | \([]\) | \(819000\) | \(1.6698\) | \(\Gamma_0(N)\)-optimal |
105710.b2 | 105710a2 | \([1, 1, 0, 284917, -49856963]\) | \(2882081488391/2883584000\) | \(-2559191414472704000\) | \([]\) | \(2457000\) | \(2.2191\) |
Rank
sage: E.rank()
The elliptic curves in class 105710.b have rank \(0\).
Complex multiplication
The elliptic curves in class 105710.b do not have complex multiplication.Modular form 105710.2.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.