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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 116032a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
116032.bk3 | 116032a1 | \([0, 1, 0, -653, -6595]\) | \(4096000/37\) | \(278592832\) | \([]\) | \(36288\) | \(0.44300\) | \(\Gamma_0(N)\)-optimal |
116032.bk2 | 116032a2 | \([0, 1, 0, -4573, 113749]\) | \(1404928000/50653\) | \(381393587008\) | \([]\) | \(108864\) | \(0.99230\) | |
116032.bk1 | 116032a3 | \([0, 1, 0, -367173, 85513301]\) | \(727057727488000/37\) | \(278592832\) | \([]\) | \(326592\) | \(1.5416\) |
Rank
sage: E.rank()
The elliptic curves in class 116032a have rank \(0\).
Complex multiplication
The elliptic curves in class 116032a do not have complex multiplication.Modular form 116032.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.