Properties

Label 72.a
Number of curves $6$
Conductor $72$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 72.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72.a1 72a5 \([0, 0, 0, -3459, 78302]\) \(3065617154/9\) \(13436928\) \([2]\) \(32\) \(0.59710\)  
72.a2 72a3 \([0, 0, 0, -579, -5362]\) \(28756228/3\) \(2239488\) \([2]\) \(16\) \(0.25053\)  
72.a3 72a4 \([0, 0, 0, -219, 1190]\) \(1556068/81\) \(60466176\) \([2, 2]\) \(16\) \(0.25053\)  
72.a4 72a2 \([0, 0, 0, -39, -70]\) \(35152/9\) \(1679616\) \([2, 2]\) \(8\) \(-0.096046\)  
72.a5 72a1 \([0, 0, 0, 6, -7]\) \(2048/3\) \(-34992\) \([4]\) \(4\) \(-0.44262\) \(\Gamma_0(N)\)-optimal
72.a6 72a6 \([0, 0, 0, 141, 4718]\) \(207646/6561\) \(-9795520512\) \([2]\) \(32\) \(0.59710\)  

Rank

sage: E.rank()
 

The elliptic curves in class 72.a have rank \(0\).

Complex multiplication

The elliptic curves in class 72.a do not have complex multiplication.

Modular form 72.2.a.a

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