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Properties

Label 7752g
Number of curves $4$
Conductor $7752$
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, 81, 0]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, 81, 0]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, 81, 0]) 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 7752g have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(17\)\(1 + T\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 7752g do not have complex multiplication.

Modular form 7752.2.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);
 
\(q - q^{3} - 2 q^{5} - 4 q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + q^{17} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

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 7752g

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
7752.a4 7752g1 \([0, -1, 0, 81, 0]\) \(3628156928/2119203\) \(-33907248\) \([2]\) \(1792\) \(0.13435\) \(\Gamma_0(N)\)-optimal
7752.a3 7752g2 \([0, -1, 0, -324, 324]\) \(14738677072/8450649\) \(2163366144\) \([2, 2]\) \(3584\) \(0.48092\)  
7752.a2 7752g3 \([0, -1, 0, -3384, -74340]\) \(4186423406308/19939113\) \(20417651712\) \([2]\) \(7168\) \(0.82749\)  
7752.a1 7752g4 \([0, -1, 0, -3744, 89244]\) \(5669532745348/14282091\) \(14624861184\) \([4]\) \(7168\) \(0.82749\)