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Properties

Label 10230bg
Number of curves $4$
Conductor $10230$
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([1, 0, 0, -486675, 130612257]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -486675, 130612257]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -486675, 130612257]) 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 10230bg have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 + T\)
\(31\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 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 10230bg do not have complex multiplication.

Modular form 10230.2.a.bg

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

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 10230bg

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
10230.bf3 10230bg1 \([1, 0, 0, -486675, 130612257]\) \(12747965531857798561201/2986780262400000\) \(2986780262400000\) \([10]\) \(176000\) \(1.9599\) \(\Gamma_0(N)\)-optimal
10230.bf4 10230bg2 \([1, 0, 0, -430355, 162005025]\) \(-8814635019030000319921/6242069790000000000\) \(-6242069790000000000\) \([10]\) \(352000\) \(2.3064\)  
10230.bf1 10230bg3 \([1, 0, 0, -8731875, -9911311503]\) \(73628549562506871957390001/178215946908754500240\) \(178215946908754500240\) \([2]\) \(880000\) \(2.7646\)  
10230.bf2 10230bg4 \([1, 0, 0, -5510855, -17315148075]\) \(-18508902577171306222471921/118801759721890483665900\) \(-118801759721890483665900\) \([2]\) \(1760000\) \(3.1111\)