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Properties

Label 422142.bh
Number of curves $6$
Conductor $422142$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bh1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 422142.bh have rank \(1\).

Complex multiplication

The elliptic curves in class 422142.bh do not have complex multiplication.

Modular form 422142.2.a.bh

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} - 6 q^{11} + q^{12} - 4 q^{13} + q^{14} + q^{16} - q^{18} - q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 422142.bh

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422142.bh1 422142bh6 \([1, 0, 1, -5177169866, 143378960934332]\) \(103665426767620308239307625/5961940992\) \(882581234916261888\) \([2]\) \(246343680\) \(3.8300\) \(\Gamma_0(N)\)-optimal*
422142.bh2 422142bh5 \([1, 0, 1, -323573706, 2240267478460]\) \(25309080274342544331625/191933498523648\) \(28413046082828419203072\) \([2]\) \(123171840\) \(3.4834\) \(\Gamma_0(N)\)-optimal*
422142.bh3 422142bh4 \([1, 0, 1, -63974891, 196291544726]\) \(195607431345044517625/752875610010048\) \(111452610234254754612672\) \([2]\) \(82114560\) \(3.2807\) \(\Gamma_0(N)\)-optimal*
422142.bh4 422142bh3 \([1, 0, 1, -5911851, -170557418]\) \(154357248921765625/89242711068672\) \(13211124069820999569408\) \([2]\) \(41057280\) \(2.9341\) \(\Gamma_0(N)\)-optimal*
422142.bh5 422142bh2 \([1, 0, 1, -4200536, -3104905606]\) \(55369510069623625/3916046302812\) \(579715395801937619868\) \([2]\) \(27371520\) \(2.7314\) \(\Gamma_0(N)\)-optimal*
422142.bh6 422142bh1 \([1, 0, 1, -4126476, -3226719494]\) \(52492168638015625/293197968\) \(43403821845873552\) \([2]\) \(13685760\) \(2.3848\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 422142.bh1.