Properties

Label 178752kg
Number of curves $4$
Conductor $178752$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("kg1")
 
E.isogeny_class()
 

Elliptic curves in class 178752kg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
178752.ep3 178752kg1 \([0, -1, 0, -182737, -30005807]\) \(350104249168/2793\) \(5383678476288\) \([2]\) \(983040\) \(1.6150\) \(\Gamma_0(N)\)-optimal
178752.ep2 178752kg2 \([0, -1, 0, -186657, -28647135]\) \(93280467172/7800849\) \(60146455937089536\) \([2, 2]\) \(1966080\) \(1.9616\)  
178752.ep1 178752kg3 \([0, -1, 0, -633537, 161276865]\) \(1823652903746/328593657\) \(5067075112454455296\) \([2]\) \(3932160\) \(2.3081\)  
178752.ep4 178752kg4 \([0, -1, 0, 197503, -131678847]\) \(55251546334/517244049\) \(-7976156544473628672\) \([2]\) \(3932160\) \(2.3081\)  

Rank

sage: E.rank()
 

The elliptic curves in class 178752kg have rank \(2\).

Complex multiplication

The elliptic curves in class 178752kg do not have complex multiplication.

Modular form 178752.2.a.kg

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