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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 4410bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4410.bi7 | 4410bk1 | \([1, -1, 1, -219407, 39596631]\) | \(13619385906841/6048000\) | \(518713499808000\) | \([4]\) | \(36864\) | \(1.7821\) | \(\Gamma_0(N)\)-optimal |
| 4410.bi6 | 4410bk2 | \([1, -1, 1, -254687, 26034999]\) | \(21302308926361/8930250000\) | \(765912902060250000\) | \([2, 2]\) | \(73728\) | \(2.1286\) | |
| 4410.bi5 | 4410bk3 | \([1, -1, 1, -649382, -152913099]\) | \(353108405631241/86318776320\) | \(7403226614433054720\) | \([4]\) | \(110592\) | \(2.3314\) | |
| 4410.bi4 | 4410bk4 | \([1, -1, 1, -1921667, -1006825809]\) | \(9150443179640281/184570312500\) | \(15829879754882812500\) | \([2]\) | \(147456\) | \(2.4752\) | |
| 4410.bi8 | 4410bk5 | \([1, -1, 1, 847813, 190527999]\) | \(785793873833639/637994920500\) | \(-54718349548988380500\) | \([2]\) | \(147456\) | \(2.4752\) | |
| 4410.bi2 | 4410bk6 | \([1, -1, 1, -9681062, -11590632651]\) | \(1169975873419524361/108425318400\) | \(9299218977357926400\) | \([2, 2]\) | \(221184\) | \(2.6780\) | |
| 4410.bi1 | 4410bk7 | \([1, -1, 1, -154893542, -741951322059]\) | \(4791901410190533590281/41160000\) | \(3530133540360000\) | \([2]\) | \(442368\) | \(3.0245\) | |
| 4410.bi3 | 4410bk8 | \([1, -1, 1, -8975462, -13352374731]\) | \(-932348627918877961/358766164249920\) | \(-30769982253764512960320\) | \([2]\) | \(442368\) | \(3.0245\) |
Rank
sage: E.rank()
The elliptic curves in class 4410bk have rank \(1\).
Complex multiplication
The elliptic curves in class 4410bk do not have complex multiplication.Modular form 4410.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
