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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12992.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12992.f1 | 12992u2 | \([0, 1, 0, -9889, 375231]\) | \(408023180713/1421\) | \(372506624\) | \([2]\) | \(12288\) | \(0.86409\) | |
12992.f2 | 12992u1 | \([0, 1, 0, -609, 5887]\) | \(-95443993/5887\) | \(-1543241728\) | \([2]\) | \(6144\) | \(0.51751\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12992.f have rank \(1\).
Complex multiplication
The elliptic curves in class 12992.f do not have complex multiplication.Modular form 12992.2.a.f
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.