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Properties

Label 42978.k
Number of curves $2$
Conductor $42978$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 42978.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42978.k1 42978m1 \([1, 1, 1, -58, -157]\) \(21601086625/4899492\) \(4899492\) \([2]\) \(12288\) \(-0.0035968\) \(\Gamma_0(N)\)-optimal
42978.k2 42978m2 \([1, 1, 1, 132, -765]\) \(254237645375/437473062\) \(-437473062\) \([2]\) \(24576\) \(0.34298\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42978.k have rank \(1\).

Complex multiplication

The elliptic curves in class 42978.k do not have complex multiplication.

Modular form 42978.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + 4 q^{7} + q^{8} + q^{9} + 4 q^{11} - q^{12} + q^{13} + 4 q^{14} + q^{16} - 6 q^{17} + q^{18} + q^{19} + O(q^{20})\) 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 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.