Properties

Label 42042.i
Number of curves $4$
Conductor $42042$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 42042.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42042.i1 42042h4 \([1, 1, 0, -3193600, 2195132416]\) \(30618029936661765625/3678951124992\) \(432824920904183808\) \([2]\) \(995328\) \(2.4088\)  
42042.i2 42042h3 \([1, 1, 0, -183040, 40173568]\) \(-5764706497797625/2612665516032\) \(-307377485295648768\) \([2]\) \(497664\) \(2.0622\)  
42042.i3 42042h2 \([1, 1, 0, -88225, -5783483]\) \(645532578015625/252306960048\) \(29683661542687152\) \([2]\) \(331776\) \(1.8595\)  
42042.i4 42042h1 \([1, 1, 0, 17615, -639659]\) \(5137417856375/4510142208\) \(-530613720628992\) \([2]\) \(165888\) \(1.5129\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 42042.i have rank \(0\).

Complex multiplication

The elliptic curves in class 42042.i do not have complex multiplication.

Modular form 42042.2.a.i

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.