Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -2861584, 1864147264]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -2861584, 1864147264]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -2861584, 1864147264]) 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 7728.b have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1 - T\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 7728.b do not have complex multiplication.

Modular form 7728.2.a.b

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);
 
\(q - q^{3} - 2 q^{5} + q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 6 q^{17} + O(q^{20})\) 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 7728.b

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
7728.b1 7728l4 \([0, -1, 0, -2861584, 1864147264]\) \(632678989847546725777/80515134\) \(329789988864\) \([4]\) \(92160\) \(2.0701\)  
7728.b2 7728l3 \([0, -1, 0, -204624, 20243520]\) \(231331938231569617/90942310746882\) \(372499704819228672\) \([2]\) \(92160\) \(2.0701\)  
7728.b3 7728l2 \([0, -1, 0, -178864, 29166784]\) \(154502321244119857/55101928644\) \(225697499725824\) \([2, 2]\) \(46080\) \(1.7236\)  
7728.b4 7728l1 \([0, -1, 0, -9584, 592320]\) \(-23771111713777/22848457968\) \(-93587283836928\) \([2]\) \(23040\) \(1.3770\) \(\Gamma_0(N)\)-optimal