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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 16928.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
16928.d1 | 16928e2 | \([0, 0, 0, -5819, -170338]\) | \(287496\) | \(75794375168\) | \([2]\) | \(12672\) | \(0.95036\) | \(-16\) | |
16928.d2 | 16928e3 | \([0, 0, 0, -5819, 170338]\) | \(287496\) | \(75794375168\) | \([2]\) | \(12672\) | \(0.95036\) | \(-16\) | |
16928.d3 | 16928e1 | \([0, 0, 0, -529, 0]\) | \(1728\) | \(9474296896\) | \([2, 2]\) | \(6336\) | \(0.60379\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
16928.d4 | 16928e4 | \([0, 0, 0, 2116, 0]\) | \(1728\) | \(-606355001344\) | \([2]\) | \(12672\) | \(0.95036\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 16928.d have rank \(1\).
Complex multiplication
Each elliptic curve in class 16928.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 16928.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.