Show commands:
SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 171600.el
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 171600.el1 | 171600bg8 | \([0, 1, 0, -418800008, 1854543359988]\) | \(126929854754212758768001/50235797102795981820\) | \(3215091014578942836480000000\) | \([2]\) | \(95551488\) | \(3.9765\) | |
| 171600.el2 | 171600bg6 | \([0, 1, 0, -365560008, 2689240079988]\) | \(84415028961834287121601/30783551683856400\) | \(1970147307766809600000000\) | \([2, 2]\) | \(47775744\) | \(3.6300\) | |
| 171600.el3 | 171600bg3 | \([0, 1, 0, -365528008, 2689734607988]\) | \(84392862605474684114881/11228954880\) | \(718653112320000000\) | \([2]\) | \(23887872\) | \(3.2834\) | |
| 171600.el4 | 171600bg7 | \([0, 1, 0, -312832008, 3492287519988]\) | \(-52902632853833942200321/51713453577420277500\) | \(-3309661028954897760000000000\) | \([2]\) | \(95551488\) | \(3.9765\) | |
| 171600.el5 | 171600bg5 | \([0, 1, 0, -188760008, -998153520012]\) | \(11621808143080380273601/1335706803288000\) | \(85485235410432000000000\) | \([2]\) | \(31850496\) | \(3.4272\) | |
| 171600.el6 | 171600bg2 | \([0, 1, 0, -12760008, -12905520012]\) | \(3590017885052913601/954068544000000\) | \(61060386816000000000000\) | \([2, 2]\) | \(15925248\) | \(3.0806\) | |
| 171600.el7 | 171600bg1 | \([0, 1, 0, -4568008, 3593167988]\) | \(164711681450297281/8097103872000\) | \(518214647808000000000\) | \([2]\) | \(7962624\) | \(2.7341\) | \(\Gamma_0(N)\)-optimal |
| 171600.el8 | 171600bg4 | \([0, 1, 0, 32167992, -83442480012]\) | \(57519563401957999679/80296734375000000\) | \(-5138991000000000000000000\) | \([2]\) | \(31850496\) | \(3.4272\) |
Rank
sage: E.rank()
The elliptic curves in class 171600.el have rank \(0\).
Complex multiplication
The elliptic curves in class 171600.el do not have complex multiplication.Modular form 171600.2.a.el
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
