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Properties

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

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Show commands: SageMath
E = EllipticCurve("cg1")
 
E.isogeny_class()
 

Elliptic curves in class 42042.cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42042.cg1 42042cn1 \([1, 1, 1, -282945650, -1833968895361]\) \(-21293376668673906679951249/26211168887701209984\) \(-3083717808469159653407616\) \([]\) \(13335840\) \(3.6088\) \(\Gamma_0(N)\)-optimal
42042.cg2 42042cn2 \([1, 1, 1, 801301360, 115092054497279]\) \(483641001192506212470106511/48918776756543177755473774\) \(-5755245166630548319753734037326\) \([]\) \(93350880\) \(4.5817\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 42042.2.a.cg

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.