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Properties

Label 139638bd
Number of curves $6$
Conductor $139638$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 139638bd have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(17\)\(1 - T\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 4 T + 5 T^{2}\) 1.5.ae
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 139638bd do not have complex multiplication.

Modular form 139638.2.a.bd

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 139638bd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139638.o5 139638bd1 \([1, 0, 1, -46575, 3584098]\) \(4354703137/352512\) \(904449347889408\) \([2]\) \(829440\) \(1.6128\) \(\Gamma_0(N)\)-optimal
139638.o4 139638bd2 \([1, 0, 1, -156095, -19590334]\) \(163936758817/30338064\) \(77839172002732176\) \([2, 2]\) \(1658880\) \(1.9594\)  
139638.o6 139638bd3 \([1, 0, 1, 309365, -113985622]\) \(1276229915423/2927177028\) \(-7510335404557732452\) \([2]\) \(3317760\) \(2.3060\)  
139638.o2 139638bd4 \([1, 0, 1, -2373875, -1407920614]\) \(576615941610337/27060804\) \(69430619471572836\) \([2, 2]\) \(3317760\) \(2.3060\)  
139638.o3 139638bd5 \([1, 0, 1, -2250665, -1560553162]\) \(-491411892194497/125563633938\) \(-322161931604735268642\) \([2]\) \(6635520\) \(2.6525\)  
139638.o1 139638bd6 \([1, 0, 1, -37981565, -90099554866]\) \(2361739090258884097/5202\) \(13346908779618\) \([2]\) \(6635520\) \(2.6525\)