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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 186184.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
186184.a1 | 186184c1 | \([0, 1, 0, -5932, 69312]\) | \(35152/17\) | \(11166041331968\) | \([2]\) | \(405504\) | \(1.1973\) | \(\Gamma_0(N)\)-optimal |
186184.a2 | 186184c2 | \([0, 1, 0, 21448, 551200]\) | \(415292/289\) | \(-759290810573824\) | \([2]\) | \(811008\) | \(1.5438\) |
Rank
sage: E.rank()
The elliptic curves in class 186184.a have rank \(1\).
Complex multiplication
The elliptic curves in class 186184.a do not have complex multiplication.Modular form 186184.2.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 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.