Show commands:
SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 50160bo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50160.bc7 | 50160bo1 | \([0, -1, 0, -1384120, -549587600]\) | \(71595431380957421881/9522562500000000\) | \(39004416000000000000\) | \([2]\) | \(1548288\) | \(2.4869\) | \(\Gamma_0(N)\)-optimal |
50160.bc5 | 50160bo2 | \([0, -1, 0, -21384120, -38053587600]\) | \(264020672568758737421881/5803468580250000\) | \(23771007304704000000\) | \([2, 2]\) | \(3096576\) | \(2.8335\) | |
50160.bc4 | 50160bo3 | \([0, -1, 0, -27934120, 56763852400]\) | \(588530213343917460371881/861551575695360000\) | \(3528915254048194560000\) | \([2]\) | \(4644864\) | \(3.0362\) | |
50160.bc6 | 50160bo4 | \([0, -1, 0, -20624120, -40884435600]\) | \(-236859095231405581781881/39282983014374049500\) | \(-160903098426876106752000\) | \([2]\) | \(6193152\) | \(3.1801\) | |
50160.bc2 | 50160bo5 | \([0, -1, 0, -342144120, -2435798739600]\) | \(1081411559614045490773061881/522522049500\) | \(2140250314752000\) | \([2]\) | \(6193152\) | \(3.1801\) | |
50160.bc3 | 50160bo6 | \([0, -1, 0, -36126120, 20758374000]\) | \(1272998045160051207059881/691293848290254950400\) | \(2831539602596884276838400\) | \([2, 2]\) | \(9289728\) | \(3.3828\) | |
50160.bc8 | 50160bo7 | \([0, -1, 0, 139464280, 163197306480]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-185119849615107546387578880\) | \([2]\) | \(18579456\) | \(3.7294\) | |
50160.bc1 | 50160bo8 | \([0, -1, 0, -342788520, -2426162248080]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(34751511809875696181575680\) | \([2]\) | \(18579456\) | \(3.7294\) |
Rank
sage: E.rank()
The elliptic curves in class 50160bo have rank \(0\).
Complex multiplication
The elliptic curves in class 50160bo do not have complex multiplication.Modular form 50160.2.a.bo
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.