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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 69696.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.y1 | 69696dh2 | \([0, 0, 0, -3625644, 2729316656]\) | \(-128667913/4096\) | \(-167791083893267890176\) | \([]\) | \(2433024\) | \(2.6568\) | |
69696.y2 | 69696dh1 | \([0, 0, 0, 207636, 13821104]\) | \(24167/16\) | \(-655433921458077696\) | \([]\) | \(811008\) | \(2.1075\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 69696.y have rank \(1\).
Complex multiplication
The elliptic curves in class 69696.y do not have complex multiplication.Modular form 69696.2.a.y
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.