Properties

Label 42042.q
Number of curves $4$
Conductor $42042$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 42042.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42042.q1 42042t4 \([1, 1, 0, -298447559, -1984618689675]\) \(24988464356366680777409257/19820013127858944\) \(2331804724479476902656\) \([2]\) \(11796480\) \(3.4062\)  
42042.q2 42042t2 \([1, 1, 0, -18779079, -30575019915]\) \(6225272619854317474537/171699142176866304\) \(20200232377966143799296\) \([2, 2]\) \(5898240\) \(3.0596\)  
42042.q3 42042t1 \([1, 1, 0, -2722759, 1046296693]\) \(18974193623767438057/6951907079749632\) \(817884916025464455168\) \([2]\) \(2949120\) \(2.7131\) \(\Gamma_0(N)\)-optimal
42042.q4 42042t3 \([1, 1, 0, 3988281, -99974486667]\) \(59633809076653006103/36736390236271934208\) \(-4321999574907156787636992\) \([2]\) \(11796480\) \(3.4062\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 42042.2.a.q

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{8} + q^{9} - 2 q^{10} + q^{11} - q^{12} - q^{13} - 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.