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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 81144p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
81144.bm2 | 81144p1 | \([0, 0, 0, -1160859, 481410790]\) | \(1969910093092/7889\) | \(692847542854656\) | \([2]\) | \(552960\) | \(2.0591\) | \(\Gamma_0(N)\)-optimal |
81144.bm1 | 81144p2 | \([0, 0, 0, -1178499, 466025182]\) | \(1030541881826/62236321\) | \(10931748531160762368\) | \([2]\) | \(1105920\) | \(2.4056\) |
Rank
sage: E.rank()
The elliptic curves in class 81144p have rank \(1\).
Complex multiplication
The elliptic curves in class 81144p do not have complex multiplication.Modular form 81144.2.a.p
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.