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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 194810.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
194810.e1 | 194810l2 | \([1, 0, 1, -3039523, 1482739806]\) | \(1753007192038126081/478174101507200\) | \(847114589440196739200\) | \([2]\) | \(12544000\) | \(2.7235\) | |
194810.e2 | 194810l1 | \([1, 0, 1, -1103523, -427704994]\) | \(83890194895342081/3958384640000\) | \(7012519851223040000\) | \([2]\) | \(6272000\) | \(2.3769\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 194810.e have rank \(0\).
Complex multiplication
The elliptic curves in class 194810.e do not have complex multiplication.Modular form 194810.2.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.