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Properties

Label 23232bw
Number of curves $6$
Conductor $23232$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bw1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 23232bw have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(11\)\(1\)
 
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.ac
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 23232bw do not have complex multiplication.

Modular form 23232.2.a.bw

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{5} + q^{9} - 2 q^{13} + 2 q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 23232bw

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
23232.do5 23232bw1 \([0, 1, 0, 323, -2653]\) \(2048/3\) \(-5442235392\) \([2]\) \(10240\) \(0.55360\) \(\Gamma_0(N)\)-optimal
23232.do4 23232bw2 \([0, 1, 0, -2097, -28305]\) \(35152/9\) \(261227298816\) \([2, 2]\) \(20480\) \(0.90017\)  
23232.do3 23232bw3 \([0, 1, 0, -11777, 465375]\) \(1556068/81\) \(9404182757376\) \([2, 2]\) \(40960\) \(1.2467\)  
23232.do2 23232bw4 \([0, 1, 0, -31137, -2124993]\) \(28756228/3\) \(348303065088\) \([2]\) \(40960\) \(1.2467\)  
23232.do6 23232bw5 \([0, 1, 0, 7583, 1863167]\) \(207646/6561\) \(-1523477606694912\) \([2]\) \(81920\) \(1.5933\)  
23232.do1 23232bw6 \([0, 1, 0, -186017, 30817983]\) \(3065617154/9\) \(2089818390528\) \([2]\) \(81920\) \(1.5933\)