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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 32192p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32192.p2 | 32192p1 | \([0, 0, 0, 500, -1008]\) | \(52734375/32192\) | \(-8438939648\) | \([2]\) | \(12096\) | \(0.59387\) | \(\Gamma_0(N)\)-optimal |
32192.p1 | 32192p2 | \([0, 0, 0, -2060, -8176]\) | \(3687953625/2024072\) | \(530598330368\) | \([2]\) | \(24192\) | \(0.94044\) |
Rank
sage: E.rank()
The elliptic curves in class 32192p have rank \(0\).
Complex multiplication
The elliptic curves in class 32192p do not have complex multiplication.Modular form 32192.2.a.p
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.