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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 82368eq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82368.be5 | 82368eq1 | \([0, 0, 0, -13836, 898576]\) | \(-1532808577/938223\) | \(-179297207451648\) | \([2]\) | \(262144\) | \(1.4370\) | \(\Gamma_0(N)\)-optimal |
| 82368.be4 | 82368eq2 | \([0, 0, 0, -247116, 47274640]\) | \(8732907467857/1656369\) | \(316537045254144\) | \([2, 2]\) | \(524288\) | \(1.7835\) | |
| 82368.be3 | 82368eq3 | \([0, 0, 0, -273036, 36751120]\) | \(11779205551777/3763454409\) | \(719207337600221184\) | \([2, 2]\) | \(1048576\) | \(2.1301\) | |
| 82368.be1 | 82368eq4 | \([0, 0, 0, -3953676, 3025866256]\) | \(35765103905346817/1287\) | \(245949530112\) | \([2]\) | \(1048576\) | \(2.1301\) | |
| 82368.be6 | 82368eq5 | \([0, 0, 0, 772404, 250439056]\) | \(266679605718863/296110251723\) | \(-56587550328374427648\) | \([2]\) | \(2097152\) | \(2.4767\) | |
| 82368.be2 | 82368eq6 | \([0, 0, 0, -1733196, -850442096]\) | \(3013001140430737/108679952667\) | \(20769062386202836992\) | \([2]\) | \(2097152\) | \(2.4767\) |
Rank
sage: E.rank()
The elliptic curves in class 82368eq have rank \(2\).
Complex multiplication
The elliptic curves in class 82368eq do not have complex multiplication.Modular form 82368.2.a.eq
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.
