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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 18810.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18810.b1 | 18810e1 | \([1, -1, 0, -33075, 1203061]\) | \(5489125095409201/2330634240000\) | \(1699032360960000\) | \([2]\) | \(129024\) | \(1.6191\) | \(\Gamma_0(N)\)-optimal |
18810.b2 | 18810e2 | \([1, -1, 0, 110925, 8777461]\) | \(207053365326094799/165767088643200\) | \(-120844207620892800\) | \([2]\) | \(258048\) | \(1.9657\) |
Rank
sage: E.rank()
The elliptic curves in class 18810.b have rank \(1\).
Complex multiplication
The elliptic curves in class 18810.b do not have complex multiplication.Modular form 18810.2.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.