Properties

Label 177600bp
Number of curves $6$
Conductor $177600$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177600.hd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177600bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177600.hd5 177600bp1 [0, 1, 0, -1352033, -2083475937] [2] 7962624 \(\Gamma_0(N)\)-optimal
177600.hd4 177600bp2 [0, 1, 0, -34120033, -76565139937] [2, 2] 15925248  
177600.hd3 177600bp3 [0, 1, 0, -46920033, -13909139937] [2, 2] 31850496  
177600.hd1 177600bp4 [0, 1, 0, -545608033, -4905523347937] [2] 31850496  
177600.hd2 177600bp5 [0, 1, 0, -485000033, 4093090860063] [4] 63700992  
177600.hd6 177600bp6 [0, 1, 0, 186359967, -110720339937] [2] 63700992  

Rank

sage: E.rank()
 

The elliptic curves in class 177600bp have rank \(0\).

Modular form 177600.2.a.hd

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.