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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 4830bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4830.bk3 | 4830bk1 | \([1, 0, 0, -480, 230400]\) | \(-12232183057921/22933241856000\) | \(-22933241856000\) | \([6]\) | \(20736\) | \(1.2426\) | \(\Gamma_0(N)\)-optimal |
| 4830.bk2 | 4830bk2 | \([1, 0, 0, -59360, 5494272]\) | \(23131609187144855041/322060536000000\) | \(322060536000000\) | \([6]\) | \(41472\) | \(1.5892\) | |
| 4830.bk4 | 4830bk3 | \([1, 0, 0, 4320, -6219840]\) | \(8915971454369279/16719623332762560\) | \(-16719623332762560\) | \([2]\) | \(62208\) | \(1.7919\) | |
| 4830.bk1 | 4830bk4 | \([1, 0, 0, -482360, -126235128]\) | \(12411881707829361287041/303132494474220600\) | \(303132494474220600\) | \([2]\) | \(124416\) | \(2.1385\) |
Rank
sage: E.rank()
The elliptic curves in class 4830bk have rank \(1\).
Complex multiplication
The elliptic curves in class 4830bk do not have complex multiplication.Modular form 4830.2.a.bk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
