Properties

Label 33a
Number of curves 44
Conductor 3333
CM no
Rank 00
Graph

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

Elliptic curves in class 33a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33.a2 33a1 [1,1,0,11,0][1, 1, 0, -11, 0] 169112377/88209169112377/88209 8820988209 [2,2][2, 2] 33 0.35921-0.35921 Γ0(N)\Gamma_0(N)-optimal
33.a3 33a2 [1,1,0,6,9][1, 1, 0, -6, -9] 30664297/29730664297/297 297297 [2][2] 66 0.70578-0.70578  
33.a1 33a3 [1,1,0,146,621][1, 1, 0, -146, 621] 347873904937/395307347873904937/395307 395307395307 [4][4] 66 0.012632-0.012632  
33.a4 33a4 [1,1,0,44,55][1, 1, 0, 44, 55] 9090072503/58458519090072503/5845851 5845851-5845851 [2][2] 66 0.012632-0.012632  

Rank

sage: E.rank()
 

The elliptic curves in class 33a have rank 00.

Complex multiplication

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

Modular form 33.2.a.a

sage: E.q_eigenform(10)
 
q+q2q3q42q5q6+4q73q8+q92q10+q11+q122q13+4q14+2q15q162q17+q18+O(q20)q + q^{2} - q^{3} - q^{4} - 2 q^{5} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9} - 2 q^{10} + q^{11} + q^{12} - 2 q^{13} + 4 q^{14} + 2 q^{15} - q^{16} - 2 q^{17} + q^{18} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the Cremona numbering.

(1222214424142441)\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with Cremona labels.