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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 123981h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123981.e5 | 123981h1 | \([1, 1, 1, -6942, 316458]\) | \(-1532808577/938223\) | \(-22646422399887\) | \([2]\) | \(294912\) | \(1.2646\) | \(\Gamma_0(N)\)-optimal |
| 123981.e4 | 123981h2 | \([1, 1, 1, -123987, 16749576]\) | \(8732907467857/1656369\) | \(39980721026961\) | \([2, 2]\) | \(589824\) | \(1.6111\) | |
| 123981.e3 | 123981h3 | \([1, 1, 1, -136992, 13004136]\) | \(11779205551777/3763454409\) | \(90840640475591721\) | \([2, 2]\) | \(1179648\) | \(1.9577\) | |
| 123981.e1 | 123981h4 | \([1, 1, 1, -1983702, 1074555468]\) | \(35765103905346817/1287\) | \(31065051303\) | \([2]\) | \(1179648\) | \(1.9577\) | |
| 123981.e6 | 123981h5 | \([1, 1, 1, 387543, 89166618]\) | \(266679605718863/296110251723\) | \(-7147381632571281387\) | \([2]\) | \(2359296\) | \(2.3043\) | |
| 123981.e2 | 123981h6 | \([1, 1, 1, -869607, -302606406]\) | \(3013001140430737/108679952667\) | \(2623269856416446523\) | \([2]\) | \(2359296\) | \(2.3043\) |
Rank
sage: E.rank()
The elliptic curves in class 123981h have rank \(1\).
Complex multiplication
The elliptic curves in class 123981h do not have complex multiplication.Modular form 123981.2.a.h
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.
