Properties

Label 235200.qt
Number of curves $6$
Conductor $235200$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("235200.qt1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 235200.qt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
235200.qt1 235200qt5 [0, 1, 0, -1317121633, 18398252484863] [2] 56623104  
235200.qt2 235200qt3 [0, 1, 0, -82321633, 287440884863] [2, 2] 28311552  
235200.qt3 235200qt6 [0, 1, 0, -76833633, 327420964863] [2] 56623104  
235200.qt4 235200qt4 [0, 1, 0, -29009633, -56851147137] [2] 28311552  
235200.qt5 235200qt2 [0, 1, 0, -5489633, 3853972863] [2, 2] 14155776  
235200.qt6 235200qt1 [0, 1, 0, 782367, 373012863] [2] 7077888 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 235200.qt have rank \(1\).

Modular form 235200.2.a.qt

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.