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Properties

Label 40656bk
Number of curves $6$
Conductor $40656$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([0, -1, 0, 1896, -7632]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 40656bk have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1 - T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 3 T + 23 T^{2}\) 1.23.d
\(29\) \( 1 - T + 29 T^{2}\) 1.29.ab
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 40656bk do not have complex multiplication.

Modular form 40656.2.a.bk

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} + 2 q^{13} + 2 q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 40656bk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40656.l6 40656bk1 \([0, -1, 0, 1896, -7632]\) \(103823/63\) \(-457147772928\) \([2]\) \(40960\) \(0.92658\) \(\Gamma_0(N)\)-optimal
40656.l5 40656bk2 \([0, -1, 0, -7784, -54096]\) \(7189057/3969\) \(28800309694464\) \([2, 2]\) \(81920\) \(1.2732\)  
40656.l3 40656bk3 \([0, -1, 0, -75544, 7968688]\) \(6570725617/45927\) \(333260726464512\) \([2]\) \(163840\) \(1.6197\)  
40656.l2 40656bk4 \([0, -1, 0, -94904, -11205456]\) \(13027640977/21609\) \(156801686114304\) \([2, 2]\) \(163840\) \(1.6197\)  
40656.l4 40656bk5 \([0, -1, 0, -65864, -18221520]\) \(-4354703137/17294403\) \(-125493616120147968\) \([2]\) \(327680\) \(1.9663\)  
40656.l1 40656bk6 \([0, -1, 0, -1517864, -719270352]\) \(53297461115137/147\) \(1066678136832\) \([2]\) \(327680\) \(1.9663\)