Properties

Label 428400nu
Number of curves $6$
Conductor $428400$
CM no
Rank $1$
Graph

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Show commands: 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)
 
\(q + q^{7} + 4 q^{11} + 2 q^{13} + q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.