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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 20800.dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20800.dk1 | 20800de3 | \([0, -1, 0, -332033, 70795937]\) | \(988345570681/44994560\) | \(184297717760000000\) | \([2]\) | \(331776\) | \(2.0749\) | |
20800.dk2 | 20800de1 | \([0, -1, 0, -52033, -4524063]\) | \(3803721481/26000\) | \(106496000000000\) | \([2]\) | \(110592\) | \(1.5256\) | \(\Gamma_0(N)\)-optimal |
20800.dk3 | 20800de2 | \([0, -1, 0, -20033, -10060063]\) | \(-217081801/10562500\) | \(-43264000000000000\) | \([2]\) | \(221184\) | \(1.8722\) | |
20800.dk4 | 20800de4 | \([0, -1, 0, 179967, 268939937]\) | \(157376536199/7722894400\) | \(-31632975462400000000\) | \([2]\) | \(663552\) | \(2.4215\) |
Rank
sage: E.rank()
The elliptic curves in class 20800.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 20800.dk do not have complex multiplication.Modular form 20800.2.a.dk
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.