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Properties

Label 3025g
Number of curves $3$
Conductor $3025$
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, 1, -1008, -29606]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 1, -1008, -29606]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 1, -1008, -29606]) 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 3025g have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(5\)\(1\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(3\) \( 1 - 3 T + 3 T^{2}\) 1.3.ad
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 3025g do not have complex multiplication.

Modular form 3025.2.a.g

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

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

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 3025g

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
3025.a3 3025g1 \([0, 1, 1, -1008, -29606]\) \(-4096/11\) \(-304487046875\) \([]\) \(3360\) \(0.89094\) \(\Gamma_0(N)\)-optimal
3025.a2 3025g2 \([0, 1, 1, -31258, 3842394]\) \(-122023936/161051\) \(-4457994853296875\) \([]\) \(16800\) \(1.6957\)  
3025.a1 3025g3 \([0, 1, 1, -23656508, 44278891894]\) \(-52893159101157376/11\) \(-304487046875\) \([]\) \(84000\) \(2.5004\)