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Properties

Label 156702cl
Number of curves $4$
Conductor $156702$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cl1")
 
E.isogeny_class()
 

Elliptic curves in class 156702cl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
156702.b3 156702cl1 \([1, 1, 0, -11491, 469309]\) \(1426487591593/179088\) \(21069524112\) \([2]\) \(374784\) \(1.0033\) \(\Gamma_0(N)\)-optimal
156702.b2 156702cl2 \([1, 1, 0, -12471, 383265]\) \(1823449422313/501132996\) \(58957795846404\) \([2, 2]\) \(749568\) \(1.3499\)  
156702.b4 156702cl3 \([1, 1, 0, 32119, 2550339]\) \(31145864569847/41657368662\) \(-4900947765715638\) \([2]\) \(1499136\) \(1.6965\)  
156702.b1 156702cl4 \([1, 1, 0, -72741, -7271025]\) \(361811696411593/16869440406\) \(1984672794325494\) \([2]\) \(1499136\) \(1.6965\)  

Rank

sage: E.rank()
 

The elliptic curves in class 156702cl have rank \(1\).

Complex multiplication

The elliptic curves in class 156702cl do not have complex multiplication.

Modular form 156702.2.a.cl

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.