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Properties

Label 465690.bq
Number of curves $4$
Conductor $465690$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bq1")
 
E.isogeny_class()
 

Elliptic curves in class 465690.bq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
465690.bq1 465690bq4 \([1, 1, 1, -302706, -63792231]\) \(65202655558249/512820150\) \(24126075751302150\) \([2]\) \(5308416\) \(1.9717\)  
465690.bq2 465690bq2 \([1, 1, 1, -31956, 537969]\) \(76711450249/41602500\) \(1957226264302500\) \([2, 2]\) \(2654208\) \(1.6251\)  
465690.bq3 465690bq1 \([1, 1, 1, -24736, 1485233]\) \(35578826569/51600\) \(2427567459600\) \([2]\) \(1327104\) \(1.2786\) \(\Gamma_0(N)\)-optimal*
465690.bq4 465690bq3 \([1, 1, 1, 123274, 4387673]\) \(4403686064471/2721093750\) \(-128016252752343750\) \([2]\) \(5308416\) \(1.9717\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 465690.bq1.

Rank

sage: E.rank()
 

The elliptic curves in class 465690.bq have rank \(1\).

Complex multiplication

The elliptic curves in class 465690.bq do not have complex multiplication.

Modular form 465690.2.a.bq

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 4 q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + 2 q^{13} + 4 q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.