Properties

Label 2142.h
Number of curves $6$
Conductor $2142$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 2142.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2142.h1 2142g5 \([1, -1, 0, -123467436, 528082919464]\) \(285531136548675601769470657/17941034271597192\) \(13079013983994352968\) \([2]\) \(245760\) \(3.1284\)  
2142.h2 2142g3 \([1, -1, 0, -7731396, 8219774992]\) \(70108386184777836280897/552468975892674624\) \(402749883425759800896\) \([2, 2]\) \(122880\) \(2.7819\)  
2142.h3 2142g6 \([1, -1, 0, -2633436, 18893883640]\) \(-2770540998624539614657/209924951154647363208\) \(-153035289391737927778632\) \([2]\) \(245760\) \(3.1284\)  
2142.h4 2142g2 \([1, -1, 0, -816516, -71166128]\) \(82582985847542515777/44772582831427584\) \(32639212884110708736\) \([2, 2]\) \(61440\) \(2.4353\)  
2142.h5 2142g1 \([1, -1, 0, -632196, -193075376]\) \(38331145780597164097/55468445663232\) \(40436496888496128\) \([2]\) \(30720\) \(2.0887\) \(\Gamma_0(N)\)-optimal
2142.h6 2142g4 \([1, -1, 0, 3149244, -562127216]\) \(4738217997934888496063/2928751705237796928\) \(-2135059993118353960512\) \([2]\) \(122880\) \(2.7819\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2142.h have rank \(1\).

Complex multiplication

The elliptic curves in class 2142.h do not have complex multiplication.

Modular form 2142.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} - q^{14} + q^{16} - q^{17} + 4 q^{19} + O(q^{20})\) 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 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.