Show commands:
SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 78144y
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78144.cu4 | 78144y1 | \([0, 1, 0, -1458337, 584100287]\) | \(1308451928740468777/194033737531392\) | \(50864780091429224448\) | \([2]\) | \(2949120\) | \(2.5059\) | \(\Gamma_0(N)\)-optimal |
| 78144.cu2 | 78144y2 | \([0, 1, 0, -22429857, 40878778815]\) | \(4760617885089919932457/133756441657344\) | \(35063448641822785536\) | \([2, 2]\) | \(5898240\) | \(2.8525\) | |
| 78144.cu3 | 78144y3 | \([0, 1, 0, -21528737, 44314749375]\) | \(-4209586785160189454377/801182513521564416\) | \(-210025188824596982267904\) | \([2]\) | \(11796480\) | \(3.1990\) | |
| 78144.cu1 | 78144y4 | \([0, 1, 0, -358875297, 2616637778367]\) | \(19499096390516434897995817/15393430272\) | \(4035295385223168\) | \([2]\) | \(11796480\) | \(3.1990\) |
Rank
sage: E.rank()
The elliptic curves in class 78144y have rank \(0\).
Complex multiplication
The elliptic curves in class 78144y do not have complex multiplication.Modular form 78144.2.a.y
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
