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Properties

Label 304200cq
Number of curves $6$
Conductor $304200$
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, 0, 0, -353455050, -2768221922375]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -353455050, -2768221922375]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -353455050, -2768221922375]) 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 304200cq have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 304200cq do not have complex multiplication.

Modular form 304200.2.a.cq

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);
 
\(q - 4 q^{11} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

Copy content 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 304200cq

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
304200.cq5 304200cq1 \([0, 0, 0, -353455050, -2768221922375]\) \(-5551350318708736/550618236675\) \(-484371121248244408668750000\) \([2]\) \(123863040\) \(3.8609\) \(\Gamma_0(N)\)-optimal
304200.cq4 304200cq2 \([0, 0, 0, -5783615175, -169294942475750]\) \(1520107298839022416/13013105625\) \(183159136916811022500000000\) \([2, 2]\) \(247726080\) \(4.2075\)  
304200.cq3 304200cq3 \([0, 0, 0, -5912139675, -161376933604250]\) \(405929061432816484/35083409765625\) \(1975192467638391056250000000000\) \([2, 2]\) \(495452160\) \(4.5541\)  
304200.cq1 304200cq4 \([0, 0, 0, -92537652675, -10834923066763250]\) \(1556580279686303289604/114075\) \(6422411112577200000000\) \([2]\) \(495452160\) \(4.5541\)  
304200.cq2 304200cq5 \([0, 0, 0, -20326656675, 931286698546750]\) \(8248670337458940482/1446075439453125\) \(162827805776352539062500000000000\) \([2]\) \(990904320\) \(4.9006\)  
304200.cq6 304200cq6 \([0, 0, 0, 6445985325, -747287997979250]\) \(263059523447441758/2294739983908125\) \(-258387264047806567710660000000000\) \([2]\) \(990904320\) \(4.9006\)