Properties

Label 242550nk
Number of curves $8$
Conductor $242550$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("nk1")
 
E.isogeny_class()
 

Elliptic curves in class 242550nk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
242550.nk8 242550nk1 \([1, -1, 1, 17518495, 3529368497]\) \(443688652450511/260789760000\) \(-349483220495640000000000\) \([2]\) \(31850496\) \(3.2069\) \(\Gamma_0(N)\)-optimal
242550.nk7 242550nk2 \([1, -1, 1, -70681505, 28401768497]\) \(29141055407581489/16604321025600\) \(22251378221944589025000000\) \([2, 2]\) \(63700992\) \(3.5535\)  
242550.nk6 242550nk3 \([1, -1, 1, -223267505, -1409736275503]\) \(-918468938249433649/109183593750000\) \(-146316458012145996093750000\) \([2]\) \(95551488\) \(3.7563\)  
242550.nk4 242550nk4 \([1, -1, 1, -826996505, 9135946998497]\) \(46676570542430835889/106752955783320\) \(143059170669060514089375000\) \([2]\) \(127401984\) \(3.9001\)  
242550.nk5 242550nk5 \([1, -1, 1, -725566505, -7488368261503]\) \(31522423139920199089/164434491947880\) \(220357945827740652710625000\) \([2]\) \(127401984\) \(3.9001\)  
242550.nk3 242550nk6 \([1, -1, 1, -3668580005, -85523595650503]\) \(4074571110566294433649/48828650062500\) \(65435061086359883789062500\) \([2, 2]\) \(191102976\) \(4.1028\)  
242550.nk2 242550nk7 \([1, -1, 1, -3765048755, -80788330588003]\) \(4404531606962679693649/444872222400201750\) \(596171325873665839295496093750\) \([2]\) \(382205952\) \(4.4494\)  
242550.nk1 242550nk8 \([1, -1, 1, -58697111255, -5473587204463003]\) \(16689299266861680229173649/2396798250\) \(3211938886282628906250\) \([2]\) \(382205952\) \(4.4494\)  

Rank

sage: E.rank()
 

The elliptic curves in class 242550nk have rank \(1\).

Complex multiplication

The elliptic curves in class 242550nk do not have complex multiplication.

Modular form 242550.2.a.nk

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} + q^{11} + 2 q^{13} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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

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

The vertices are labelled with Cremona labels.