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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 10830a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10830.a2 | 10830a1 | \([1, 1, 0, -527428, -204088112]\) | \(-50284268371/26542080\) | \(-8564802689466040320\) | \([2]\) | \(194560\) | \(2.3370\) | \(\Gamma_0(N)\)-optimal |
10830.a1 | 10830a2 | \([1, 1, 0, -9306948, -10930905648]\) | \(276288773643091/41990400\) | \(13549785504819321600\) | \([2]\) | \(389120\) | \(2.6836\) |
Rank
sage: E.rank()
The elliptic curves in class 10830a have rank \(1\).
Complex multiplication
The elliptic curves in class 10830a do not have complex multiplication.Modular form 10830.2.a.a
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.