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Properties

Label 2142j
Number of curves $3$
Conductor $2142$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 2142j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2142.i3 2142j1 \([1, -1, 0, 972, -314928]\) \(139233463487/58763045376\) \(-42838260079104\) \([]\) \(8640\) \(1.2948\) \(\Gamma_0(N)\)-optimal
2142.i2 2142j2 \([1, -1, 0, -8748, 8508888]\) \(-101566487155393/42823570577256\) \(-31218382950819624\) \([3]\) \(25920\) \(1.8441\)  
2142.i1 2142j3 \([1, -1, 0, -3435318, 2451690342]\) \(-6150311179917589675873/244053849830826\) \(-177915256526672154\) \([3]\) \(77760\) \(2.3934\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 2142.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 3 q^{5} + q^{7} - q^{8} - 3 q^{10} - 3 q^{11} + 5 q^{13} - q^{14} + q^{16} + q^{17} + 2 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.