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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 69696.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
69696.x1 | 69696bf2 | \([0, 0, 0, -31944, 2254714]\) | \(-32768\) | \(-110012407471296\) | \([]\) | \(190080\) | \(1.4718\) | \(-11\) | |
69696.x2 | 69696bf1 | \([0, 0, 0, -264, -1694]\) | \(-32768\) | \(-62099136\) | \([]\) | \(17280\) | \(0.27281\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 69696.x have rank \(0\).
Complex multiplication
Each elliptic curve in class 69696.x has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 69696.2.a.x
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.