Properties

Label 33.a
Number of curves $4$
Conductor $33$
CM no
Rank $0$
Graph

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

Elliptic curves in class 33.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33.a1 33a3 \([1, 1, 0, -146, 621]\) \(347873904937/395307\) \(395307\) \([4]\) \(6\) \(-0.012632\)  
33.a2 33a1 \([1, 1, 0, -11, 0]\) \(169112377/88209\) \(88209\) \([2, 2]\) \(3\) \(-0.35921\) \(\Gamma_0(N)\)-optimal
33.a3 33a2 \([1, 1, 0, -6, -9]\) \(30664297/297\) \(297\) \([2]\) \(6\) \(-0.70578\)  
33.a4 33a4 \([1, 1, 0, 44, 55]\) \(9090072503/5845851\) \(-5845851\) \([2]\) \(6\) \(-0.012632\)  

Rank

sage: E.rank()
 

The elliptic curves in class 33.a have rank \(0\).

Complex multiplication

The elliptic curves in class 33.a do not have complex multiplication.

Modular form 33.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} + 4q^{7} - 3q^{8} + q^{9} - 2q^{10} + q^{11} + q^{12} - 2q^{13} + 4q^{14} + 2q^{15} - q^{16} - 2q^{17} + q^{18} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.