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Properties

Label 2800v
Number of curves $6$
Conductor $2800$
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([0, 1, 0, -208, -2412]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -208, -2412]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -208, -2412]) 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 2800v have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(5\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 7 T + 17 T^{2}\) 1.17.h
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 2800v do not have complex multiplication.

Modular form 2800.2.a.v

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);
 
\(q - 2 q^{3} + q^{7} + q^{9} + 4 q^{13} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 2800v

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
2800.g5 2800v1 \([0, 1, 0, -208, -2412]\) \(-15625/28\) \(-1792000000\) \([2]\) \(1152\) \(0.46578\) \(\Gamma_0(N)\)-optimal
2800.g4 2800v2 \([0, 1, 0, -4208, -106412]\) \(128787625/98\) \(6272000000\) \([2]\) \(2304\) \(0.81236\)  
2800.g6 2800v3 \([0, 1, 0, 1792, 49588]\) \(9938375/21952\) \(-1404928000000\) \([2]\) \(3456\) \(1.0151\)  
2800.g3 2800v4 \([0, 1, 0, -14208, 529588]\) \(4956477625/941192\) \(60236288000000\) \([2]\) \(6912\) \(1.3617\)  
2800.g2 2800v5 \([0, 1, 0, -68208, 6853588]\) \(-548347731625/1835008\) \(-117440512000000\) \([2]\) \(10368\) \(1.5644\)  
2800.g1 2800v6 \([0, 1, 0, -1092208, 438981588]\) \(2251439055699625/25088\) \(1605632000000\) \([2]\) \(20736\) \(1.9110\)