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Properties

Label 221760kz
Number of curves $8$
Conductor $221760$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("kz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 221760kz have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 7 T + 17 T^{2}\) 1.17.ah
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 5 T + 29 T^{2}\) 1.29.f
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 221760kz do not have complex multiplication.

Modular form 221760.2.a.kz

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} + q^{7} - q^{11} - 2 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 221760kz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221760.dy6 221760kz1 \([0, 0, 0, -165043308, -816103020112]\) \(2601656892010848045529/56330588160\) \(10764943037206364160\) \([2]\) \(21233664\) \(3.1788\) \(\Gamma_0(N)\)-optimal
221760.dy5 221760kz2 \([0, 0, 0, -165227628, -814188820048]\) \(2610383204210122997209/12104550027662400\) \(2313215533427166963302400\) \([2, 2]\) \(42467328\) \(3.5254\)  
221760.dy4 221760kz3 \([0, 0, 0, -176111148, -700405336528]\) \(3160944030998056790089/720291785342976000\) \(137649903767395894296576000\) \([2]\) \(63700992\) \(3.7281\)  
221760.dy3 221760kz4 \([0, 0, 0, -252157548, 135329310128]\) \(9278380528613437145689/5328033205714065000\) \(1018203001838778026557440000\) \([2]\) \(84934656\) \(3.8720\)  
221760.dy7 221760kz5 \([0, 0, 0, -81246828, -1641198146128]\) \(-310366976336070130009/5909282337130963560\) \(-1129281440649962438063554560\) \([2]\) \(84934656\) \(3.8720\)  
221760.dy2 221760kz6 \([0, 0, 0, -931085868, 10336419100208]\) \(467116778179943012100169/28800309694464000000\) \(5503824892333721124864000000\) \([2, 2]\) \(127401984\) \(4.0747\)  
221760.dy1 221760kz7 \([0, 0, 0, -14669561388, 683867929163312]\) \(1826870018430810435423307849/7641104625000000000\) \(1460237833764864000000000000\) \([2]\) \(254803968\) \(4.4213\)  
221760.dy8 221760kz8 \([0, 0, 0, 727794132, 43161672988208]\) \(223090928422700449019831/4340371122724101696000\) \(-829457838497037061032247296000\) \([2]\) \(254803968\) \(4.4213\)