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Properties

Label 233730cx
Number of curves $4$
Conductor $233730$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, 1546357, -2313006069]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, 1546357, -2313006069]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, 1546357, -2313006069]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 233730cx have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(7\)\(1\)
\(53\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 233730cx do not have complex multiplication.

Modular form 233730.2.a.cx

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + 2 q^{13} + q^{16} + 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 233730cx

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
233730.cx4 233730cx1 \([1, -1, 1, 1546357, -2313006069]\) \(4768013769464231/29697948831600\) \(-2547077872942814223600\) \([2]\) \(11796480\) \(2.7893\) \(\Gamma_0(N)\)-optimal
233730.cx3 233730cx2 \([1, -1, 1, -19630463, -30376527933]\) \(9754377335041367449/995626517602500\) \(85391024379504644902500\) \([2, 2]\) \(23592960\) \(3.1358\)  
233730.cx2 233730cx3 \([1, -1, 1, -72140333, 202620267231]\) \(484108118865316036729/73399966614843750\) \(6295230418084649458593750\) \([2]\) \(47185920\) \(3.4824\)  
233730.cx1 233730cx4 \([1, -1, 1, -305949713, -2059692844233]\) \(36928196050908253259449/452758954469850\) \(38831379272894645951850\) \([2]\) \(47185920\) \(3.4824\)