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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 122199.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122199.e1 | 122199b2 | \([1, 1, 1, -713103, -232076688]\) | \(270902819202625/1004157\) | \(148651274190573\) | \([2]\) | \(1148928\) | \(1.9352\) | |
122199.e2 | 122199b1 | \([1, 1, 1, -43918, -3750766]\) | \(-63282696625/4032567\) | \(-596964640797063\) | \([2]\) | \(574464\) | \(1.5887\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 122199.e have rank \(0\).
Complex multiplication
The elliptic curves in class 122199.e do not have complex multiplication.Modular form 122199.2.a.e
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.