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Properties

Label 10470.a
Number of curves $4$
Conductor $10470$
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([1, 1, 0, -232937, 43157829]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -232937, 43157829]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -232937, 43157829]) 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 10470.a have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(349\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 10470.a do not have complex multiplication.

Modular form 10470.2.a.a

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} - 2 q^{13} + 4 q^{14} - q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 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 10470.a

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
10470.a1 10470a4 \([1, 1, 0, -232937, 43157829]\) \(1397790417785316543001/640892891563200\) \(640892891563200\) \([4]\) \(124416\) \(1.7981\)  
10470.a2 10470a3 \([1, 1, 0, -128617, -17502779]\) \(235301185027835613721/4636822725000000\) \(4636822725000000\) \([2]\) \(124416\) \(1.7981\)  
10470.a3 10470a2 \([1, 1, 0, -16937, 433029]\) \(537369779909439001/227309898240000\) \(227309898240000\) \([2, 2]\) \(62208\) \(1.4516\)  
10470.a4 10470a1 \([1, 1, 0, 3543, 52101]\) \(4916382075769319/3952292659200\) \(-3952292659200\) \([2]\) \(31104\) \(1.1050\) \(\Gamma_0(N)\)-optimal