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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 169050ii
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.w4 | 169050ii1 | \([1, 1, 0, 353265475, -4644711079875]\) | \(2652277923951208297919/6605028468326400000\) | \(-12141796785470822400000000000\) | \([2]\) | \(176947200\) | \(4.0727\) | \(\Gamma_0(N)\)-optimal |
| 169050.w3 | 169050ii2 | \([1, 1, 0, -2964622525, -51888118311875]\) | \(1567558142704512417614401/274462175610000000000\) | \(504534382786576406250000000000\) | \([2, 2]\) | \(353894400\) | \(4.4193\) | |
| 169050.w2 | 169050ii3 | \([1, 1, 0, -13788330525, 574685514100125]\) | \(157706830105239346386477121/13650704956054687500000\) | \(25093621677732467651367187500000\) | \([2]\) | \(707788800\) | \(4.7659\) | |
| 169050.w1 | 169050ii4 | \([1, 1, 0, -45227122525, -3701973455811875]\) | \(5565604209893236690185614401/229307220930246900000\) | \(421527581800353399032812500000\) | \([2]\) | \(707788800\) | \(4.7659\) |
Rank
sage: E.rank()
The elliptic curves in class 169050ii have rank \(0\).
Complex multiplication
The elliptic curves in class 169050ii do not have complex multiplication.Modular form 169050.2.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.
