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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 161700bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 161700.dg3 | 161700bt1 | \([0, 1, 0, -3260133, -2300347512]\) | \(-130287139815424/2250652635\) | \(-66196757963778750000\) | \([2]\) | \(5971968\) | \(2.6012\) | \(\Gamma_0(N)\)-optimal |
| 161700.dg2 | 161700bt2 | \([0, 1, 0, -52376508, -145916628012]\) | \(33766427105425744/9823275\) | \(4622793921900000000\) | \([2]\) | \(11943936\) | \(2.9478\) | |
| 161700.dg4 | 161700bt3 | \([0, 1, 0, 12615867, -11014287012]\) | \(7549996227362816/6152409907875\) | \(-180956218312896468750000\) | \([2]\) | \(17915904\) | \(3.1505\) | |
| 161700.dg1 | 161700bt4 | \([0, 1, 0, -60755508, -96125082012]\) | \(52702650535889104/22020583921875\) | \(10362798711298687500000000\) | \([2]\) | \(35831808\) | \(3.4971\) |
Rank
sage: E.rank()
The elliptic curves in class 161700bt have rank \(0\).
Complex multiplication
The elliptic curves in class 161700bt do not have complex multiplication.Modular form 161700.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
