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Properties

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

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 42042.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42042.b1 42042v1 \([1, 1, 0, -5095339, -4429108739]\) \(-124352595912593543977/103332962304\) \(-12157019682103296\) \([]\) \(1179360\) \(2.3897\) \(\Gamma_0(N)\)-optimal
42042.b2 42042v2 \([1, 1, 0, -3955354, -6461653484]\) \(-58169016237585194137/119573538788081664\) \(-14067707264879019687936\) \([]\) \(3538080\) \(2.9391\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 42042.2.a.b

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.