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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 2112z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2112.p1 | 2112z1 | \([0, 1, 0, -32065, 2199359]\) | \(55635379958596/24057\) | \(1576599552\) | \([2]\) | \(5376\) | \(1.1073\) | \(\Gamma_0(N)\)-optimal |
2112.p2 | 2112z2 | \([0, 1, 0, -31905, 2222559]\) | \(-27403349188178/578739249\) | \(-75856510844928\) | \([2]\) | \(10752\) | \(1.4539\) |
Rank
sage: E.rank()
The elliptic curves in class 2112z have rank \(1\).
Complex multiplication
The elliptic curves in class 2112z do not have complex multiplication.Modular form 2112.2.a.z
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.