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Properties

Label 47040ee
Number of curves $8$
Conductor $47040$
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, 4639, 371841]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, 4639, 371841]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, 4639, 371841]) 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 47040ee have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 47040ee do not have complex multiplication.

Modular form 47040.2.a.ee

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);
 
\(q - q^{3} - q^{5} + q^{9} + 2 q^{13} + q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

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 47040ee

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
47040.u8 47040ee1 \([0, -1, 0, 4639, 371841]\) \(357911/2160\) \(-66616515624960\) \([2]\) \(110592\) \(1.3344\) \(\Gamma_0(N)\)-optimal
47040.u6 47040ee2 \([0, -1, 0, -58081, 4900225]\) \(702595369/72900\) \(2248307402342400\) \([2, 2]\) \(221184\) \(1.6810\)  
47040.u7 47040ee3 \([0, -1, 0, -42401, -10983615]\) \(-273359449/1536000\) \(-47371744444416000\) \([2]\) \(331776\) \(1.8837\)  
47040.u5 47040ee4 \([0, -1, 0, -214881, -32951295]\) \(35578826569/5314410\) \(163901609630760960\) \([2]\) \(442368\) \(2.0275\)  
47040.u4 47040ee5 \([0, -1, 0, -904801, 331564801]\) \(2656166199049/33750\) \(1040883056640000\) \([2]\) \(442368\) \(2.0275\)  
47040.u3 47040ee6 \([0, -1, 0, -1045921, -410585279]\) \(4102915888729/9000000\) \(277568815104000000\) \([2, 2]\) \(663552\) \(2.2303\)  
47040.u1 47040ee7 \([0, -1, 0, -16725921, -26323353279]\) \(16778985534208729/81000\) \(2498119335936000\) \([2]\) \(1327104\) \(2.5768\)  
47040.u2 47040ee8 \([0, -1, 0, -1422241, -88380095]\) \(10316097499609/5859375000\) \(180708864000000000000\) \([2]\) \(1327104\) \(2.5768\)