Properties

Label 168b
Number of curves $4$
Conductor $168$
CM no
Rank $0$
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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 168b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
168.a4 168b1 \([0, -1, 0, -7, 52]\) \(-2725888/64827\) \(-1037232\) \([4]\) \(24\) \(-0.16496\) \(\Gamma_0(N)\)-optimal
168.a3 168b2 \([0, -1, 0, -252, 1620]\) \(6940769488/35721\) \(9144576\) \([2, 2]\) \(48\) \(0.18162\)  
168.a2 168b3 \([0, -1, 0, -392, -228]\) \(6522128932/3720087\) \(3809369088\) \([2]\) \(96\) \(0.52819\)  
168.a1 168b4 \([0, -1, 0, -4032, 99900]\) \(7080974546692/189\) \(193536\) \([2]\) \(96\) \(0.52819\)  

Rank

sage: E.rank()
 

The elliptic curves in class 168b have rank \(0\).

Complex multiplication

The elliptic curves in class 168b do not have complex multiplication.

Modular form 168.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + q^{7} + q^{9} + 6 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\)  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.