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SageMath
This isogeny class and its quadratic twist by $\Q(\sqrt{-3})$ are the ones of minimal conductor with a $13$-isogeny.
sage: E = EllipticCurve("b1")
sage: E.isogeny_class()
Elliptic curves in class 147.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
147.b1 | 147c2 | [0, -1, 1, -912, 10919] | [] | 78 | |
147.b2 | 147c1 | [0, -1, 1, -2, -1] | [] | 6 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 147.b have rank \(0\).
Complex multiplication
The elliptic curves in class 147.b do not have complex multiplication.Modular form 147.2.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.
\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.