Properties

Label 163254.b
Number of curves $4$
Conductor $163254$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 163254.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
163254.b1 163254cv4 \([1, 1, 0, -26332786886, -1644736185372756]\) \(418363773366867122598323253553/8175657796488\) \(39462338633008446792\) \([2]\) \(173408256\) \(4.2224\)  
163254.b2 163254cv3 \([1, 1, 0, -1672077046, -24836473343300]\) \(107110600388332385155666993/6780210942198563598456\) \(32726783197702506564099806904\) \([2]\) \(173408256\) \(4.2224\)  
163254.b3 163254cv2 \([1, 1, 0, -1645800926, -25699459952460]\) \(102139918202985795140993713/451521457781599296\) \(2179407836113343516326464\) \([2, 2]\) \(86704128\) \(3.8758\)  
163254.b4 163254cv1 \([1, 1, 0, -101222046, -415012602636]\) \(-23762325430118066146993/1660616206139584512\) \(-8015477249340401778782208\) \([2]\) \(43352064\) \(3.5292\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 163254.b have rank \(1\).

Complex multiplication

The elliptic curves in class 163254.b do not have complex multiplication.

Modular form 163254.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + 2 q^{10} - q^{12} + q^{14} + 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.