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Properties

Label 221760ek
Number of curves $8$
Conductor $221760$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ek1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 221760ek have rank \(0\).

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 \(\mathbb{F}_p\)
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 221760.2.a.ek

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 221760ek

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221760.cb8 221760ek1 \([0, 0, 0, 915252, 42198928]\) \(443688652450511/260789760000\) \(-49837699246325760000\) \([2]\) \(5308416\) \(2.4690\) \(\Gamma_0(N)\)-optimal
221760.cb7 221760ek2 \([0, 0, 0, -3692748, 338954128]\) \(29141055407581489/16604321025600\) \(3173135162451532185600\) \([2, 2]\) \(10616832\) \(2.8156\)  
221760.cb6 221760ek3 \([0, 0, 0, -11664588, -16835301488]\) \(-918468938249433649/109183593750000\) \(-20865309696000000000000\) \([2]\) \(15925248\) \(3.0183\)  
221760.cb4 221760ek4 \([0, 0, 0, -43206348, 109096186768]\) \(46676570542430835889/106752955783320\) \(20400807546988863160320\) \([2]\) \(21233664\) \(3.1621\)  
221760.cb5 221760ek5 \([0, 0, 0, -37907148, -89425945712]\) \(31522423139920199089/164434491947880\) \(31423920768287904890880\) \([2]\) \(21233664\) \(3.1621\)  
221760.cb3 221760ek6 \([0, 0, 0, -191664588, -1021307301488]\) \(4074571110566294433649/48828650062500\) \(9331300341006336000000\) \([2, 2]\) \(31850496\) \(3.3649\)  
221760.cb2 221760ek7 \([0, 0, 0, -196704588, -964760517488]\) \(4404531606962679693649/444872222400201750\) \(85016405640412417425408000\) \([2]\) \(63700992\) \(3.7114\)  
221760.cb1 221760ek8 \([0, 0, 0, -3066624588, -65364062085488]\) \(16689299266861680229173649/2396798250\) \(458035278446592000\) \([2]\) \(63700992\) \(3.7114\)