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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 428400.ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
428400.ek1 | 428400ek3 | \([0, 0, 0, -18284475, 30093430250]\) | \(14489843500598257/6246072\) | \(291416735232000000\) | \([2]\) | \(18874368\) | \(2.6937\) | \(\Gamma_0(N)\)-optimal* |
428400.ek2 | 428400ek4 | \([0, 0, 0, -2444475, -777577750]\) | \(34623662831857/14438442312\) | \(673639964508672000000\) | \([2]\) | \(18874368\) | \(2.6937\) | |
428400.ek3 | 428400ek2 | \([0, 0, 0, -1148475, 465286250]\) | \(3590714269297/73410624\) | \(3425046073344000000\) | \([2, 2]\) | \(9437184\) | \(2.3471\) | \(\Gamma_0(N)\)-optimal* |
428400.ek4 | 428400ek1 | \([0, 0, 0, 3525, 21766250]\) | \(103823/4386816\) | \(-204671287296000000\) | \([2]\) | \(4718592\) | \(2.0006\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 428400.ek have rank \(0\).
Complex multiplication
The elliptic curves in class 428400.ek do not have complex multiplication.Modular form 428400.2.a.ek
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.