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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 85680dy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85680.q3 | 85680dy1 | \([0, 0, 0, -2591643, 1606618442]\) | \(-644706081631626841/347004000000\) | \(-1036148391936000000\) | \([2]\) | \(1769472\) | \(2.4060\) | \(\Gamma_0(N)\)-optimal |
85680.q2 | 85680dy2 | \([0, 0, 0, -41471643, 102795706442]\) | \(2641739317048851306841/764694000\) | \(2283364048896000\) | \([2]\) | \(3538944\) | \(2.7526\) | |
85680.q4 | 85680dy3 | \([0, 0, 0, 2106357, 6525683642]\) | \(346124368852751159/6361262220902400\) | \(-18994627211419031961600\) | \([2]\) | \(5308416\) | \(2.9553\) | |
85680.q1 | 85680dy4 | \([0, 0, 0, -42130443, 99361032122]\) | \(2769646315294225853641/174474906948464640\) | \(520979280549604239605760\) | \([2]\) | \(10616832\) | \(3.3019\) |
Rank
sage: E.rank()
The elliptic curves in class 85680dy have rank \(0\).
Complex multiplication
The elliptic curves in class 85680dy do not have complex multiplication.Modular form 85680.2.a.dy
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.