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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 221778k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221778.w1 | 221778k1 | \([1, -1, 1, -8471, 329779]\) | \(-35937/4\) | \(-7481658208644\) | \([]\) | \(623376\) | \(1.2077\) | \(\Gamma_0(N)\)-optimal |
221778.w2 | 221778k2 | \([1, -1, 1, 53134, -483407]\) | \(109503/64\) | \(-9696229038402624\) | \([]\) | \(1870128\) | \(1.7570\) |
Rank
sage: E.rank()
The elliptic curves in class 221778k have rank \(0\).
Complex multiplication
The elliptic curves in class 221778k do not have complex multiplication.Modular form 221778.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.