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SageMath
E = EllipticCurve("nu1")
E.isogeny_class()
Elliptic curves in class 428400nu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
428400.nu5 | 428400nu1 | \([0, 0, 0, -252878475, 1545867400250]\) | \(38331145780597164097/55468445663232\) | \(2587935800863752192000000\) | \([2]\) | \(94371840\) | \(3.5866\) | \(\Gamma_0(N)\)-optimal |
428400.nu4 | 428400nu2 | \([0, 0, 0, -326606475, 570962056250]\) | \(82582985847542515777/44772582831427584\) | \(2088909624583085359104000000\) | \([2, 2]\) | \(188743680\) | \(3.9332\) | |
428400.nu6 | 428400nu3 | \([0, 0, 0, 1259697525, 4490719240250]\) | \(4738217997934888496063/2928751705237796928\) | \(-136643839559574653472768000000\) | \([2]\) | \(377487360\) | \(4.2797\) | |
428400.nu2 | 428400nu4 | \([0, 0, 0, -3092558475, -65742737143750]\) | \(70108386184777836280897/552468975892674624\) | \(25775992539248627257344000000\) | \([2, 2]\) | \(377487360\) | \(4.2797\) | |
428400.nu3 | 428400nu5 | \([0, 0, 0, -1053374475, -151145802247750]\) | \(-2770540998624539614657/209924951154647363208\) | \(-9794258521071227377832448000000\) | \([2]\) | \(754974720\) | \(4.6263\) | |
428400.nu1 | 428400nu6 | \([0, 0, 0, -49386974475, -4224416420839750]\) | \(285531136548675601769470657/17941034271597192\) | \(837056894975638589952000000\) | \([2]\) | \(754974720\) | \(4.6263\) |
Rank
sage: E.rank()
The elliptic curves in class 428400nu have rank \(1\).
Complex multiplication
The elliptic curves in class 428400nu do not have complex multiplication.Modular form 428400.2.a.nu
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.