Properties

Label 2142d
Number of curves $4$
Conductor $2142$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2142d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2142.g4 2142d1 \([1, -1, 0, 9, -2723]\) \(103823/4386816\) \(-3197988864\) \([2]\) \(1536\) \(0.50271\) \(\Gamma_0(N)\)-optimal
2142.g3 2142d2 \([1, -1, 0, -2871, -57443]\) \(3590714269297/73410624\) \(53516344896\) \([2, 2]\) \(3072\) \(0.84928\)  
2142.g1 2142d3 \([1, -1, 0, -45711, -3750251]\) \(14489843500598257/6246072\) \(4553386488\) \([2]\) \(6144\) \(1.1959\)  
2142.g2 2142d4 \([1, -1, 0, -6111, 98725]\) \(34623662831857/14438442312\) \(10525624445448\) \([2]\) \(6144\) \(1.1959\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2142d have rank \(0\).

Complex multiplication

The elliptic curves in class 2142d do not have complex multiplication.

Modular form 2142.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 2q^{5} - q^{7} - q^{8} - 2q^{10} - 6q^{13} + q^{14} + q^{16} - q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.