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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 210210be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
210210.eh4 | 210210be1 | \([1, 0, 0, 3184, 90096]\) | \(30342134159/47190000\) | \(-5551856310000\) | \([2]\) | \(589824\) | \(1.1307\) | \(\Gamma_0(N)\)-optimal |
210210.eh3 | 210210be2 | \([1, 0, 0, -21316, 908396]\) | \(9104453457841/2226896100\) | \(261992099268900\) | \([2, 2]\) | \(1179648\) | \(1.4773\) | |
210210.eh1 | 210210be3 | \([1, 0, 0, -317766, 68914026]\) | \(30161840495801041/2799263610\) | \(329330564452890\) | \([2]\) | \(2359296\) | \(1.8238\) | |
210210.eh2 | 210210be4 | \([1, 0, 0, -116866, -14628034]\) | \(1500376464746641/83599963590\) | \(9835452116399910\) | \([2]\) | \(2359296\) | \(1.8238\) |
Rank
sage: E.rank()
The elliptic curves in class 210210be have rank \(0\).
Complex multiplication
The elliptic curves in class 210210be do not have complex multiplication.Modular form 210210.2.a.be
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.