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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1422i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1422.f3 | 1422i1 | \([1, -1, 1, -419, -3193]\) | \(11134383337/316\) | \(230364\) | \([]\) | \(400\) | \(0.13055\) | \(\Gamma_0(N)\)-optimal |
| 1422.f2 | 1422i2 | \([1, -1, 1, -734, 2477]\) | \(59914169497/31554496\) | \(23003227584\) | \([3]\) | \(1200\) | \(0.67986\) | |
| 1422.f1 | 1422i3 | \([1, -1, 1, -46949, 3927197]\) | \(15698803397448457/20709376\) | \(15097135104\) | \([3]\) | \(3600\) | \(1.2292\) |
Rank
sage: E.rank()
The elliptic curves in class 1422i have rank \(0\).
Complex multiplication
The elliptic curves in class 1422i do not have complex multiplication.Modular form 1422.2.a.i
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
