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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 81600ii
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81600.ff3 | 81600ii1 | \([0, 1, 0, -161633, -21895137]\) | \(114013572049/15667200\) | \(64172851200000000\) | \([2]\) | \(884736\) | \(1.9518\) | \(\Gamma_0(N)\)-optimal |
| 81600.ff2 | 81600ii2 | \([0, 1, 0, -673633, 190584863]\) | \(8253429989329/936360000\) | \(3835330560000000000\) | \([2, 2]\) | \(1769472\) | \(2.2984\) | |
| 81600.ff4 | 81600ii3 | \([0, 1, 0, 926367, 960184863]\) | \(21464092074671/109596256200\) | \(-448906265395200000000\) | \([2]\) | \(3538944\) | \(2.6450\) | |
| 81600.ff1 | 81600ii4 | \([0, 1, 0, -10465633, 13027896863]\) | \(30949975477232209/478125000\) | \(1958400000000000000\) | \([2]\) | \(3538944\) | \(2.6450\) |
Rank
sage: E.rank()
The elliptic curves in class 81600ii have rank \(1\).
Complex multiplication
The elliptic curves in class 81600ii do not have complex multiplication.Modular form 81600.2.a.ii
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.
