Properties

Label 36414.j
Number of curves $6$
Conductor $36414$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 36414.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
36414.j1 36414t6 \([1, -1, 0, -35682089058, 2594328654970476]\) \(285531136548675601769470657/17941034271597192\) \(315695602490628590375454792\) \([2]\) \(70778880\) \(4.5450\)  
36414.j2 36414t4 \([1, -1, 0, -2234373498, 40374817041780]\) \(70108386184777836280897/552468975892674624\) \(9721403100931233571553461824\) \([2, 2]\) \(35389440\) \(4.1985\)  
36414.j3 36414t5 \([1, -1, 0, -761063058, 92822606071164]\) \(-2770540998624539614657/209924951154647363208\) \(-3693899857128042261673746625608\) \([2]\) \(70778880\) \(4.5450\)  
36414.j4 36414t2 \([1, -1, 0, -235973178, -350583079500]\) \(82582985847542515777/44772582831427584\) \(787831253095911235754102784\) \([2, 2]\) \(17694720\) \(3.8519\)  
36414.j5 36414t1 \([1, -1, 0, -182704698, -949310141004]\) \(38331145780597164097/55468445663232\) \(976038733764360595832832\) \([2]\) \(8847360\) \(3.5053\) \(\Gamma_0(N)\)-optimal
36414.j6 36414t3 \([1, -1, 0, 910131462, -2758090486284]\) \(4738217997934888496063/2928751705237796928\) \(-51535157903033793888281675328\) \([2]\) \(35389440\) \(4.1985\)  

Rank

sage: E.rank()
 

The elliptic curves in class 36414.j have rank \(0\).

Complex multiplication

The elliptic curves in class 36414.j do not have complex multiplication.

Modular form 36414.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} - q^{7} - q^{8} + 2 q^{10} + 4 q^{11} - 2 q^{13} + q^{14} + q^{16} + 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.