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Properties

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

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Show commands: SageMath
E = EllipticCurve("dk1")
 
E.isogeny_class()
 

Elliptic curves in class 42042dk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42042.db2 42042dk1 \([1, 0, 0, -72521, -7522971]\) \(860833894093732321/8282804244\) \(405857407956\) \([]\) \(159936\) \(1.3890\) \(\Gamma_0(N)\)-optimal
42042.db1 42042dk2 \([1, 0, 0, -2337511, 1374987977]\) \(28826282175168869972161/9077387406557184\) \(444791982921302016\) \([7]\) \(1119552\) \(2.3619\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 42042.2.a.dk

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