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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 63063t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63063.w5 | 63063t1 | \([1, -1, 0, -10593, 604624]\) | \(-1532808577/938223\) | \(-80467747342983\) | \([2]\) | \(196608\) | \(1.3702\) | \(\Gamma_0(N)\)-optimal |
| 63063.w4 | 63063t2 | \([1, -1, 0, -189198, 31717615]\) | \(8732907467857/1656369\) | \(142060344074649\) | \([2, 2]\) | \(393216\) | \(1.7168\) | |
| 63063.w3 | 63063t3 | \([1, -1, 0, -209043, 24672640]\) | \(11779205551777/3763454409\) | \(322776886220277489\) | \([2, 2]\) | \(786432\) | \(2.0634\) | |
| 63063.w1 | 63063t4 | \([1, -1, 0, -3027033, 2027850754]\) | \(35765103905346817/1287\) | \(110380997727\) | \([2]\) | \(786432\) | \(2.0634\) | |
| 63063.w6 | 63063t5 | \([1, -1, 0, 591372, 167626759]\) | \(266679605718863/296110251723\) | \(-25396227678615276483\) | \([2]\) | \(1572864\) | \(2.4099\) | |
| 63063.w2 | 63063t6 | \([1, -1, 0, -1326978, -569398019]\) | \(3013001140430737/108679952667\) | \(9321057970712194707\) | \([2]\) | \(1572864\) | \(2.4099\) |
Rank
sage: E.rank()
The elliptic curves in class 63063t have rank \(0\).
Complex multiplication
The elliptic curves in class 63063t do not have complex multiplication.Modular form 63063.2.a.t
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
