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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 212160do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212160.u2 | 212160do1 | \([0, -1, 0, -8236, 409186]\) | \(-965484891701056/556329515625\) | \(-35605089000000\) | \([2]\) | \(589824\) | \(1.3031\) | \(\Gamma_0(N)\)-optimal |
212160.u1 | 212160do2 | \([0, -1, 0, -146361, 21597561]\) | \(84653093985344704/14681057625\) | \(60133612032000\) | \([2]\) | \(1179648\) | \(1.6497\) |
Rank
sage: E.rank()
The elliptic curves in class 212160do have rank \(1\).
Complex multiplication
The elliptic curves in class 212160do do not have complex multiplication.Modular form 212160.2.a.do
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.