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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 6864.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6864.e1 | 6864r3 | \([0, -1, 0, -109824, -13972032]\) | \(35765103905346817/1287\) | \(5271552\) | \([2]\) | \(16384\) | \(1.2342\) | |
| 6864.e2 | 6864r5 | \([0, -1, 0, -48144, 3953280]\) | \(3013001140430737/108679952667\) | \(445153086124032\) | \([4]\) | \(32768\) | \(1.5808\) | |
| 6864.e3 | 6864r4 | \([0, -1, 0, -7584, -167616]\) | \(11779205551777/3763454409\) | \(15415109259264\) | \([2, 4]\) | \(16384\) | \(1.2342\) | |
| 6864.e4 | 6864r2 | \([0, -1, 0, -6864, -216576]\) | \(8732907467857/1656369\) | \(6784487424\) | \([2, 2]\) | \(8192\) | \(0.88767\) | |
| 6864.e5 | 6864r1 | \([0, -1, 0, -384, -4032]\) | \(-1532808577/938223\) | \(-3842961408\) | \([2]\) | \(4096\) | \(0.54110\) | \(\Gamma_0(N)\)-optimal |
| 6864.e6 | 6864r6 | \([0, -1, 0, 21456, -1166592]\) | \(266679605718863/296110251723\) | \(-1212867591057408\) | \([4]\) | \(32768\) | \(1.5808\) |
Rank
sage: E.rank()
The elliptic curves in class 6864.e have rank \(0\).
Complex multiplication
The elliptic curves in class 6864.e do not have complex multiplication.Modular form 6864.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
