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Properties

Label 4200d
Number of curves $6$
Conductor $4200$
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, -383, 12012]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -383, 12012]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -383, 12012]) 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 4200d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 5 T + 11 T^{2}\) 1.11.f
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 9 T + 23 T^{2}\) 1.23.j
\(29\) \( 1 + 5 T + 29 T^{2}\) 1.29.f
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 4200d do not have complex multiplication.

Modular form 4200.2.a.d

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);
 
\(q - q^{3} + q^{7} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 4200d

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
4200.n5 4200d1 \([0, -1, 0, -383, 12012]\) \(-24918016/229635\) \(-57408750000\) \([2]\) \(3072\) \(0.74691\) \(\Gamma_0(N)\)-optimal
4200.n4 4200d2 \([0, -1, 0, -10508, 417012]\) \(32082281296/99225\) \(396900000000\) \([2, 2]\) \(6144\) \(1.0935\)  
4200.n3 4200d3 \([0, -1, 0, -15008, 30012]\) \(23366901604/13505625\) \(216090000000000\) \([2, 2]\) \(12288\) \(1.4401\)  
4200.n1 4200d4 \([0, -1, 0, -168008, 26562012]\) \(32779037733124/315\) \(5040000000\) \([2]\) \(12288\) \(1.4401\)  
4200.n2 4200d5 \([0, -1, 0, -162008, -24959988]\) \(14695548366242/57421875\) \(1837500000000000\) \([2]\) \(24576\) \(1.7866\)  
4200.n6 4200d6 \([0, -1, 0, 59992, 180012]\) \(746185003198/432360075\) \(-13835522400000000\) \([2]\) \(24576\) \(1.7866\)