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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 173400.ei
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
173400.ei1 | 173400cu2 | \([0, 1, 0, -445409208, 3618004223088]\) | \(101215672859338/2601\) | \(251127267876000000000\) | \([2]\) | \(30965760\) | \(3.4305\) | |
173400.ei2 | 173400cu1 | \([0, 1, 0, -27804208, 56668783088]\) | \(-49241558516/250563\) | \(-12095963402694000000000\) | \([2]\) | \(15482880\) | \(3.0839\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 173400.ei have rank \(1\).
Complex multiplication
The elliptic curves in class 173400.ei do not have complex multiplication.Modular form 173400.2.a.ei
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.