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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 84150fr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
84150.hc1 | 84150fr1 | \([1, -1, 1, -203405, -34979403]\) | \(81706955619457/744505344\) | \(8480381184000000\) | \([2]\) | \(1146880\) | \(1.8786\) | \(\Gamma_0(N)\)-optimal |
84150.hc2 | 84150fr2 | \([1, -1, 1, -59405, -83651403]\) | \(-2035346265217/264305213568\) | \(-3010601573298000000\) | \([2]\) | \(2293760\) | \(2.2252\) |
Rank
sage: E.rank()
The elliptic curves in class 84150fr have rank \(0\).
Complex multiplication
The elliptic curves in class 84150fr do not have complex multiplication.Modular form 84150.2.a.fr
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.