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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 169065.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169065.w1 | 169065bk2 | \([1, -1, 0, -63345, 4414046]\) | \(43132764843/12138425\) | \(7910785916698275\) | \([2]\) | \(1179648\) | \(1.7576\) | |
169065.w2 | 169065bk1 | \([1, -1, 0, 10350, 449255]\) | \(188132517/244205\) | \(-159151906016415\) | \([2]\) | \(589824\) | \(1.4110\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169065.w have rank \(1\).
Complex multiplication
The elliptic curves in class 169065.w do not have complex multiplication.Modular form 169065.2.a.w
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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.