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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 4998.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4998.b1 | 4998i2 | \([1, 1, 0, -667356, -208043184]\) | \(814544990575471/9268826496\) | \(374030581770771072\) | \([2]\) | \(112896\) | \(2.1852\) | |
4998.b2 | 4998i1 | \([1, 1, 0, -8796, -8236080]\) | \(-1865409391/724451328\) | \(-29234224180740096\) | \([2]\) | \(56448\) | \(1.8387\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4998.b have rank \(1\).
Complex multiplication
The elliptic curves in class 4998.b do not have complex multiplication.Modular form 4998.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.