Properties

Label 33600.hf
Number of curves 8
Conductor 33600
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("33600.hf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 33600.hf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33600.hf1 33600da8 [0, 1, 0, -3073280033, -65577989759937] [2] 9437184  
33600.hf2 33600da6 [0, 1, 0, -192080033, -1024703759937] [2, 2] 4718592  
33600.hf3 33600da7 [0, 1, 0, -190880033, -1038137759937] [2] 9437184  
33600.hf4 33600da4 [0, 1, 0, -24112033, 45551504063] [2] 2359296  
33600.hf5 33600da3 [0, 1, 0, -12080033, -15803759937] [2, 2] 2359296  
33600.hf6 33600da2 [0, 1, 0, -1712033, 505104063] [2, 2] 1179648  
33600.hf7 33600da1 [0, 1, 0, 335967, 56592063] [2] 589824 \(\Gamma_0(N)\)-optimal
33600.hf8 33600da5 [0, 1, 0, 2031967, -50505167937] [2] 4718592  

Rank

sage: E.rank()
 

The elliptic curves in class 33600.hf have rank \(1\).

Modular form 33600.2.a.hf

sage: E.q_eigenform(10)
 
\( q + q^{3} + q^{7} + q^{9} + 4q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.