Properties

Label 265200.dz
Number of curves 8
Conductor 265200
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("265200.dz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 265200.dz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
265200.dz1 265200dz7 [0, 1, 0, -2636400002408, 1647649523114107188] [2] 1911029760  
265200.dz2 265200dz6 [0, 1, 0, -164775002408, 25744481864107188] [2, 2] 955514880  
265200.dz3 265200dz8 [0, 1, 0, -164476210408, 25842499384539188] [2] 1911029760  
265200.dz4 265200dz4 [0, 1, 0, -32549306408, 2259971694187188] [2] 637009920  
265200.dz5 265200dz3 [0, 1, 0, -10317114408, 400722685291188] [2] 477757440  
265200.dz6 265200dz2 [0, 1, 0, -2222906408, 28373223787188] [2, 2] 318504960  
265200.dz7 265200dz1 [0, 1, 0, -838458408, -8998565524812] [2] 159252480 \(\Gamma_0(N)\)-optimal
265200.dz8 265200dz5 [0, 1, 0, 5952325592, 188591420523188] [2] 637009920  

Rank

sage: E.rank()
 

The elliptic curves in class 265200.dz have rank \(1\).

Modular form 265200.2.a.dz

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.