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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 39984bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
39984.bn2 | 39984bt1 | \([0, -1, 0, -16872, 1188720]\) | \(-1102302937/616896\) | \(-297276200976384\) | \([2]\) | \(110592\) | \(1.4804\) | \(\Gamma_0(N)\)-optimal |
39984.bn1 | 39984bt2 | \([0, -1, 0, -299112, 63055728]\) | \(6141556990297/1019592\) | \(491331498835968\) | \([2]\) | \(221184\) | \(1.8270\) |
Rank
sage: E.rank()
The elliptic curves in class 39984bt have rank \(1\).
Complex multiplication
The elliptic curves in class 39984bt do not have complex multiplication.Modular form 39984.2.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 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.