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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1827.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1827.b1 | 1827c2 | \([1, -1, 1, -320, -2100]\) | \(4956477625/52983\) | \(38624607\) | \([2]\) | \(512\) | \(0.27157\) | |
1827.b2 | 1827c1 | \([1, -1, 1, -5, -84]\) | \(-15625/4263\) | \(-3107727\) | \([2]\) | \(256\) | \(-0.075002\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1827.b have rank \(1\).
Complex multiplication
The elliptic curves in class 1827.b do not have complex multiplication.Modular form 1827.2.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.