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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 16830.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16830.p1 | 16830c1 | \([1, -1, 0, -3255, 61325]\) | \(193802978403/31790000\) | \(625722570000\) | \([2]\) | \(36864\) | \(0.98519\) | \(\Gamma_0(N)\)-optimal |
16830.p2 | 16830c2 | \([1, -1, 0, 5925, 338561]\) | \(1168574089437/3214062500\) | \(-63262392187500\) | \([2]\) | \(73728\) | \(1.3318\) |
Rank
sage: E.rank()
The elliptic curves in class 16830.p have rank \(1\).
Complex multiplication
The elliptic curves in class 16830.p do not have complex multiplication.Modular form 16830.2.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 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.