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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 14700bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14700.z2 | 14700bm1 | \([0, 1, 0, -766033, -251348812]\) | \(4927700992/151875\) | \(1532176015781250000\) | \([2]\) | \(322560\) | \(2.2651\) | \(\Gamma_0(N)\)-optimal |
14700.z1 | 14700bm2 | \([0, 1, 0, -1837908, 606151188]\) | \(4253563312/1476225\) | \(238284013974300000000\) | \([2]\) | \(645120\) | \(2.6116\) |
Rank
sage: E.rank()
The elliptic curves in class 14700bm have rank \(0\).
Complex multiplication
The elliptic curves in class 14700bm do not have complex multiplication.Modular form 14700.2.a.bm
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 Cremona 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 Cremona labels.