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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 23232cz 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 \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(17\) \( 1 + T + 17 T^{2}\) 1.17.b
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 9 T + 29 T^{2}\) 1.29.aj
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 23232.2.a.cz

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 23232cz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
23232.bo5 23232cz1 \([0, -1, 0, 323, 2653]\) \(2048/3\) \(-5442235392\) \([2]\) \(10240\) \(0.55360\) \(\Gamma_0(N)\)-optimal
23232.bo4 23232cz2 \([0, -1, 0, -2097, 28305]\) \(35152/9\) \(261227298816\) \([2, 2]\) \(20480\) \(0.90017\)  
23232.bo3 23232cz3 \([0, -1, 0, -11777, -465375]\) \(1556068/81\) \(9404182757376\) \([2, 2]\) \(40960\) \(1.2467\)  
23232.bo2 23232cz4 \([0, -1, 0, -31137, 2124993]\) \(28756228/3\) \(348303065088\) \([2]\) \(40960\) \(1.2467\)  
23232.bo6 23232cz5 \([0, -1, 0, 7583, -1863167]\) \(207646/6561\) \(-1523477606694912\) \([2]\) \(81920\) \(1.5933\)  
23232.bo1 23232cz6 \([0, -1, 0, -186017, -30817983]\) \(3065617154/9\) \(2089818390528\) \([2]\) \(81920\) \(1.5933\)