Properties

Label 167310.bq
Number of curves $4$
Conductor $167310$
CM no
Rank $1$
Graph

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

Elliptic curves in class 167310.bq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
167310.bq1 167310dg4 \([1, -1, 0, -3455061804, 78169375649808]\) \(1296294060988412126189641/647824320\) \(2279527784224067520\) \([2]\) \(55738368\) \(3.7587\)  
167310.bq2 167310dg3 \([1, -1, 0, -215940204, 1221450744528]\) \(-316472948332146183241/7074906009600\) \(-24894781380941406105600\) \([2]\) \(27869184\) \(3.4121\)  
167310.bq3 167310dg2 \([1, -1, 0, -42736329, 106808747553]\) \(2453170411237305241/19353090685500\) \(68098567105670338165500\) \([2]\) \(18579456\) \(3.2094\)  
167310.bq4 167310dg1 \([1, -1, 0, -908829, 3837808053]\) \(-23592983745241/1794399750000\) \(-6314032925052459750000\) \([2]\) \(9289728\) \(2.8628\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 167310.bq have rank \(1\).

Complex multiplication

The elliptic curves in class 167310.bq do not have complex multiplication.

Modular form 167310.2.a.bq

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - 2 q^{7} - q^{8} - q^{10} - q^{11} + 2 q^{14} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.