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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 75726d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75726.g2 | 75726d1 | \([1, -1, 0, -803619, -326622699]\) | \(-78731237277328508209/17734929828102144\) | \(-12928763844686462976\) | \([]\) | \(1749888\) | \(2.3873\) | \(\Gamma_0(N)\)-optimal |
75726.g1 | 75726d2 | \([1, -1, 0, -34230159, 99790755921]\) | \(-6084472608417988103640049/2379033678299675392884\) | \(-1734315551480463361412436\) | \([]\) | \(12249216\) | \(3.3602\) |
Rank
sage: E.rank()
The elliptic curves in class 75726d have rank \(1\).
Complex multiplication
The elliptic curves in class 75726d do not have complex multiplication.Modular form 75726.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.