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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1815.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1815.b1 | 1815e2 | \([0, 1, 1, -724951, -237822920]\) | \(-196566176333824/421875\) | \(-90432652921875\) | \([]\) | \(14256\) | \(1.9255\) | |
1815.b2 | 1815e1 | \([0, 1, 1, -6211, -530909]\) | \(-123633664/492075\) | \(-105480646368075\) | \([3]\) | \(4752\) | \(1.3762\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1815.b have rank \(1\).
Complex multiplication
The elliptic curves in class 1815.b do not have complex multiplication.Modular form 1815.2.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.