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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 127296dr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127296.r2 | 127296dr1 | \([0, 0, 0, -516, 704]\) | \(5088448/2873\) | \(8578732032\) | \([2]\) | \(61440\) | \(0.59607\) | \(\Gamma_0(N)\)-optimal |
127296.r1 | 127296dr2 | \([0, 0, 0, -5196, -143440]\) | \(649461896/3757\) | \(89746735104\) | \([2]\) | \(122880\) | \(0.94265\) |
Rank
sage: E.rank()
The elliptic curves in class 127296dr have rank \(1\).
Complex multiplication
The elliptic curves in class 127296dr do not have complex multiplication.Modular form 127296.2.a.dr
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.