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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 270400.bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270400.bt1 | 270400bt2 | \([0, 1, 0, -22273, -1286817]\) | \(16974593\) | \(71991296000\) | \([2]\) | \(393216\) | \(1.1455\) | |
270400.bt2 | 270400bt1 | \([0, 1, 0, -1473, -18017]\) | \(4913\) | \(71991296000\) | \([2]\) | \(196608\) | \(0.79895\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 270400.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 270400.bt do not have complex multiplication.Modular form 270400.2.a.bt
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.