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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 342720bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342720.bw7 | 342720bw1 | \([0, 0, 0, -18619788, 36861334288]\) | \(-3735772816268612449/909650165760000\) | \(-173836853795629301760000\) | \([2]\) | \(28311552\) | \(3.1778\) | \(\Gamma_0(N)\)-optimal |
| 342720.bw6 | 342720bw2 | \([0, 0, 0, -313531788, 2136752739088]\) | \(17836145204788591940449/770635366502400\) | \(147270711949459351142400\) | \([2, 2]\) | \(56623104\) | \(3.5244\) | |
| 342720.bw8 | 342720bw3 | \([0, 0, 0, 133997172, -250767224048]\) | \(1392333139184610040991/947901937500000000\) | \(-181146881212416000000000000\) | \([2]\) | \(84934656\) | \(3.7271\) | |
| 342720.bw3 | 342720bw4 | \([0, 0, 0, -5016456588, 136755153384208]\) | \(73054578035931991395831649/136386452160\) | \(26063856893857628160\) | \([2]\) | \(113246208\) | \(3.8709\) | |
| 342720.bw5 | 342720bw5 | \([0, 0, 0, -329198988, 1911402001168]\) | \(20645800966247918737249/3688936444974392640\) | \(704966732909466677296496640\) | \([2]\) | \(113246208\) | \(3.8709\) | |
| 342720.bw4 | 342720bw6 | \([0, 0, 0, -586002828, -2091375224048]\) | \(116454264690812369959009/57505157319440250000\) | \(10989406699093214429184000000\) | \([2, 2]\) | \(169869312\) | \(4.0737\) | |
| 342720.bw2 | 342720bw7 | \([0, 0, 0, -5031282828, 135906118983952]\) | \(73704237235978088924479009/899277423164136103500\) | \(171854591816277745847894016000\) | \([2]\) | \(339738624\) | \(4.4202\) | |
| 342720.bw1 | 342720bw8 | \([0, 0, 0, -7660722828, -257887781432048]\) | \(260174968233082037895439009/223081361502731896500\) | \(42631512073303897551273984000\) | \([2]\) | \(339738624\) | \(4.4202\) |
Rank
sage: E.rank()
The elliptic curves in class 342720bw have rank \(0\).
Complex multiplication
The elliptic curves in class 342720bw do not have complex multiplication.Modular form 342720.2.a.bw
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.
