Properties

Label 144.b
Number of curves $6$
Conductor $144$
CM no
Rank $0$
Graph

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

Elliptic curves in class 144.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
144.b1 144b5 \([0, 0, 0, -3459, -78302]\) \(3065617154/9\) \(13436928\) \([2]\) \(64\) \(0.59710\)  
144.b2 144b4 \([0, 0, 0, -579, 5362]\) \(28756228/3\) \(2239488\) \([4]\) \(32\) \(0.25053\)  
144.b3 144b3 \([0, 0, 0, -219, -1190]\) \(1556068/81\) \(60466176\) \([2, 2]\) \(32\) \(0.25053\)  
144.b4 144b2 \([0, 0, 0, -39, 70]\) \(35152/9\) \(1679616\) \([2, 2]\) \(16\) \(-0.096046\)  
144.b5 144b1 \([0, 0, 0, 6, 7]\) \(2048/3\) \(-34992\) \([2]\) \(8\) \(-0.44262\) \(\Gamma_0(N)\)-optimal
144.b6 144b6 \([0, 0, 0, 141, -4718]\) \(207646/6561\) \(-9795520512\) \([2]\) \(64\) \(0.59710\)  

Rank

sage: E.rank()
 

The elliptic curves in class 144.b have rank \(0\).

Complex multiplication

The elliptic curves in class 144.b do not have complex multiplication.

Modular form 144.2.a.b

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.