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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 47190.bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47190.bl1 | 47190bh8 | \([1, 0, 1, -126687003, 308562320218]\) | \(126929854754212758768001/50235797102795981820\) | \(88995778951226352349021020\) | \([2]\) | \(19906560\) | \(3.6776\) | |
| 47190.bl2 | 47190bh6 | \([1, 0, 1, -110581903, 447433376498]\) | \(84415028961834287121601/30783551683856400\) | \(54534939604604327840400\) | \([2, 2]\) | \(9953280\) | \(3.3310\) | |
| 47190.bl3 | 47190bh3 | \([1, 0, 1, -110572223, 447515652626]\) | \(84392862605474684114881/11228954880\) | \(19892778536167680\) | \([2]\) | \(4976640\) | \(2.9845\) | |
| 47190.bl4 | 47190bh7 | \([1, 0, 1, -94631683, 581038799306]\) | \(-52902632853833942200321/51713453577420277500\) | \(-91613537533068244228177500\) | \([2]\) | \(19906560\) | \(3.6776\) | |
| 47190.bl5 | 47190bh5 | \([1, 0, 1, -57099903, -166062081902]\) | \(11621808143080380273601/1335706803288000\) | \(2366286080139692568000\) | \([2]\) | \(6635520\) | \(3.1283\) | |
| 47190.bl6 | 47190bh2 | \([1, 0, 1, -3859903, -2146769902]\) | \(3590017885052913601/954068544000000\) | \(1690190623877184000000\) | \([2, 2]\) | \(3317760\) | \(2.7817\) | |
| 47190.bl7 | 47190bh1 | \([1, 0, 1, -1381823, 597951506]\) | \(164711681450297281/8097103872000\) | \(14344513432584192000\) | \([2]\) | \(1658880\) | \(2.4352\) | \(\Gamma_0(N)\)-optimal |
| 47190.bl8 | 47190bh4 | \([1, 0, 1, 9730817, -13883715694]\) | \(57519563401957999679/80296734375000000\) | \(-142250563046109375000000\) | \([2]\) | \(6635520\) | \(3.1283\) |
Rank
sage: E.rank()
The elliptic curves in class 47190.bl have rank \(0\).
Complex multiplication
The elliptic curves in class 47190.bl do not have complex multiplication.Modular form 47190.2.a.bl
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.
