Properties

Label 331200hr
Number of curves $6$
Conductor $331200$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("331200.hr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 331200hr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
331200.hr5 331200hr1 [0, 0, 0, -6048300, 6281282000] [2] 14155776 \(\Gamma_0(N)\)-optimal
331200.hr4 331200hr2 [0, 0, 0, -99360300, 381208898000] [2, 2] 28311552  
331200.hr1 331200hr3 [0, 0, 0, -1589760300, 24397514498000] [2] 56623104  
331200.hr3 331200hr4 [0, 0, 0, -101952300, 360270722000] [2, 2] 56623104  
331200.hr6 331200hr5 [0, 0, 0, 126575700, 1745607458000] [2] 113246208  
331200.hr2 331200hr6 [0, 0, 0, -371952300, -2365109278000] [2] 113246208  

Rank

sage: E.rank()
 

The elliptic curves in class 331200hr have rank \(2\).

Modular form 331200.2.a.hr

sage: E.q_eigenform(10)
 
\( q - 4q^{11} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.