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Properties

Label 42042.r
Number of curves $2$
Conductor $42042$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 42042.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42042.r1 42042r2 \([1, 1, 0, -5709, 163647]\) \(174958262857/33462\) \(3936770838\) \([2]\) \(46080\) \(0.84189\)  
42042.r2 42042r1 \([1, 1, 0, -319, 3025]\) \(-30664297/18876\) \(-2220742524\) \([2]\) \(23040\) \(0.49531\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 42042.r have rank \(1\).

Complex multiplication

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

Modular form 42042.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{8} + q^{9} - 2 q^{10} + q^{11} - q^{12} - q^{13} - 2 q^{15} + q^{16} - q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.