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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 193200en
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.p4 | 193200en1 | \([0, -1, 0, -276008, -59929488]\) | \(-36333758230561/3290930160\) | \(-210619530240000000\) | \([2]\) | \(1769472\) | \(2.0671\) | \(\Gamma_0(N)\)-optimal |
| 193200.p3 | 193200en2 | \([0, -1, 0, -4508008, -3682521488]\) | \(158306179791523681/1143116100\) | \(73159430400000000\) | \([2, 2]\) | \(3538944\) | \(2.4137\) | |
| 193200.p2 | 193200en3 | \([0, -1, 0, -4600008, -3524281488]\) | \(168197522113656001/13424780328750\) | \(859185941040000000000\) | \([2]\) | \(7077888\) | \(2.7603\) | |
| 193200.p1 | 193200en4 | \([0, -1, 0, -72128008, -235754361488]\) | \(648418741232906810881/33810\) | \(2163840000000\) | \([2]\) | \(7077888\) | \(2.7603\) |
Rank
sage: E.rank()
The elliptic curves in class 193200en have rank \(1\).
Complex multiplication
The elliptic curves in class 193200en do not have complex multiplication.Modular form 193200.2.a.en
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.
