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Properties

Label 235200.mq
Number of curves $4$
Conductor $235200$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -3333633, -2340460863]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -3333633, -2340460863]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -3333633, -2340460863]) 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 235200.mq have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 235200.mq do not have complex multiplication.

Modular form 235200.2.a.mq

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);
 
\(q - q^{3} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{17} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

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

Elliptic curves in class 235200.mq

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
235200.mq1 235200mq4 \([0, -1, 0, -3333633, -2340460863]\) \(68017239368/39375\) \(2371803840000000000\) \([2]\) \(4718592\) \(2.4714\)  
235200.mq2 235200mq3 \([0, -1, 0, -1961633, 1042891137]\) \(13858588808/229635\) \(13832359994880000000\) \([2]\) \(4718592\) \(2.4714\)  
235200.mq3 235200mq2 \([0, -1, 0, -246633, -22123863]\) \(220348864/99225\) \(747118209600000000\) \([2, 2]\) \(2359296\) \(2.1249\)  
235200.mq4 235200mq1 \([0, -1, 0, 53492, -2615738]\) \(143877824/108045\) \(-12711386205000000\) \([2]\) \(1179648\) \(1.7783\) \(\Gamma_0(N)\)-optimal