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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 31680z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31680.s2 | 31680z1 | \([0, 0, 0, 72, 648]\) | \(55296/275\) | \(-205286400\) | \([2]\) | \(8192\) | \(0.27724\) | \(\Gamma_0(N)\)-optimal |
31680.s1 | 31680z2 | \([0, 0, 0, -828, 8208]\) | \(5256144/605\) | \(7226081280\) | \([2]\) | \(16384\) | \(0.62381\) |
Rank
sage: E.rank()
The elliptic curves in class 31680z have rank \(1\).
Complex multiplication
The elliptic curves in class 31680z do not have complex multiplication.Modular form 31680.2.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.