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Properties

Label 41616ci
Number of curves $6$
Conductor $41616$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ci1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 41616ci have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 41616ci do not have complex multiplication.

Modular form 41616.2.a.ci

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 41616ci

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
41616.s5 41616ci1 \([0, 0, 0, -1415811, 601361026]\) \(4354703137/352512\) \(25407089071333834752\) \([2]\) \(884736\) \(2.4664\) \(\Gamma_0(N)\)-optimal
41616.s4 41616ci2 \([0, 0, 0, -4745091, -3281245310]\) \(163936758817/30338064\) \(2186597603201668153344\) \([2, 2]\) \(1769472\) \(2.8130\)  
41616.s6 41616ci3 \([0, 0, 0, 9404349, -19108808894]\) \(1276229915423/2927177028\) \(-210974512861855069913088\) \([2]\) \(3538944\) \(3.1596\)  
41616.s2 41616ci4 \([0, 0, 0, -72163011, -235940487230]\) \(576615941610337/27060804\) \(1950391071991611408384\) \([2, 2]\) \(3538944\) \(3.1596\)  
41616.s3 41616ci5 \([0, 0, 0, -68417571, -261524089694]\) \(-491411892194497/125563633938\) \(-9049922929100632024424448\) \([2]\) \(7077888\) \(3.5061\)  
41616.s1 41616ci6 \([0, 0, 0, -1154595171, -15100548367646]\) \(2361739090258884097/5202\) \(374931001920724992\) \([2]\) \(7077888\) \(3.5061\)