Properties

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

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Show commands: SageMath
E = EllipticCurve("dz1")
 
E.isogeny_class()
 

Elliptic curves in class 265200.dz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
265200.dz1 265200dz7 \([0, 1, 0, -2636400002408, 1647649523114107188]\) \(31664865542564944883878115208137569/103216295812500\) \(6605842932000000000000\) \([2]\) \(1911029760\) \(5.2721\)  
265200.dz2 265200dz6 \([0, 1, 0, -164775002408, 25744481864107188]\) \(7730680381889320597382223137569/441370202660156250000\) \(28247692970250000000000000000\) \([2, 2]\) \(955514880\) \(4.9255\)  
265200.dz3 265200dz8 \([0, 1, 0, -164476210408, 25842499384539188]\) \(-7688701694683937879808871873249/58423707246780395507812500\) \(-3739117263793945312500000000000000\) \([2]\) \(1911029760\) \(5.2721\)  
265200.dz4 265200dz4 \([0, 1, 0, -32549306408, 2259971694187188]\) \(59589391972023341137821784609/8834417507562311995200\) \(565402720483987967692800000000\) \([2]\) \(637009920\) \(4.7228\)  
265200.dz5 265200dz3 \([0, 1, 0, -10317114408, 400722685291188]\) \(1897660325010178513043539489/14258428094958372000000\) \(912539398077335808000000000000\) \([2]\) \(477757440\) \(4.5789\)  
265200.dz6 265200dz2 \([0, 1, 0, -2222906408, 28373223787188]\) \(18980483520595353274840609/5549773448629762560000\) \(355185500712304803840000000000\) \([2, 2]\) \(318504960\) \(4.3762\)  
265200.dz7 265200dz1 \([0, 1, 0, -838458408, -8998565524812]\) \(1018563973439611524445729/42904970360310988800\) \(2745918103059903283200000000\) \([2]\) \(159252480\) \(4.0296\) \(\Gamma_0(N)\)-optimal
265200.dz8 265200dz5 \([0, 1, 0, 5952325592, 188591420523188]\) \(364421318680576777174674911/450962301637624725000000\) \(-28861587304807982400000000000000\) \([2]\) \(637009920\) \(4.7228\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 265200.dz do not have complex multiplication.

Modular form 265200.2.a.dz

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

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.