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SageMath
sage: E = EllipticCurve("bt1")
sage: E.isogeny_class()
Elliptic curves in class 121275.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 121275.bt1 | 121275en6 | [1, -1, 1, -33617430230, -2372429651687728] | [2] | 70778880 | |
| 121275.bt2 | 121275en4 | [1, -1, 1, -2101089605, -37068811375228] | [2, 2] | 35389440 | |
| 121275.bt3 | 121275en5 | [1, -1, 1, -2090670980, -37454633896228] | [2] | 70778880 | |
| 121275.bt4 | 121275en3 | [1, -1, 1, -280531355, 953779972772] | [2] | 35389440 | |
| 121275.bt5 | 121275en2 | [1, -1, 1, -131969480, -573138978478] | [2, 2] | 17694720 | |
| 121275.bt6 | 121275en1 | [1, -1, 1, 385645, -26777022478] | [2] | 8847360 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121275.bt have rank \(0\).
Complex multiplication
The elliptic curves in class 121275.bt do not have complex multiplication.Modular form 121275.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
