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Properties

Label 17640bg
Number of curves $6$
Conductor $17640$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bg1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 17640bg have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 4 T + 23 T^{2}\) 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 17640bg do not have complex multiplication.

Modular form 17640.2.a.bg

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - 4 q^{11} + 2 q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 17640bg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
17640.bp4 17640bg1 \([0, 0, 0, -324282, 71077489]\) \(2748251600896/2205\) \(3025828748880\) \([2]\) \(98304\) \(1.7003\) \(\Gamma_0(N)\)-optimal
17640.bp3 17640bg2 \([0, 0, 0, -326487, 70061866]\) \(175293437776/4862025\) \(106751238260486400\) \([2, 2]\) \(196608\) \(2.0469\)  
17640.bp2 17640bg3 \([0, 0, 0, -758667, -154412426]\) \(549871953124/200930625\) \(17646633263468160000\) \([2, 2]\) \(393216\) \(2.3934\)  
17640.bp5 17640bg4 \([0, 0, 0, 70413, 229536286]\) \(439608956/259416045\) \(-22783086494526919680\) \([2]\) \(393216\) \(2.3934\)  
17640.bp1 17640bg5 \([0, 0, 0, -10760547, -13582936514]\) \(784478485879202/221484375\) \(38903512485600000000\) \([2]\) \(786432\) \(2.7400\)  
17640.bp6 17640bg6 \([0, 0, 0, 2328333, -1092243026]\) \(7947184069438/7533176175\) \(-1323194981047023974400\) \([2]\) \(786432\) \(2.7400\)