↑

Properties

Label 7728r
Number of curves $6$
Conductor $7728$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, 2016, -70668]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, 2016, -70668]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, 2016, -70668]) 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 7728r have rank \(0\).

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 \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 - 5 T + 11 T^{2}\) 1.11.af
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 7728.2.a.r

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);
 
\(q + q^{3} - 2 q^{5} - q^{7} + q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

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 7728r

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.o5 7728r1 \([0, 1, 0, 2016, -70668]\) \(221115865823/664731648\) \(-2722740830208\) \([2]\) \(12288\) \(1.0694\) \(\Gamma_0(N)\)-optimal
7728.o4 7728r2 \([0, 1, 0, -18464, -832524]\) \(169967019783457/26337394944\) \(107877969690624\) \([2, 2]\) \(24576\) \(1.4160\)  
7728.o2 7728r3 \([0, 1, 0, -283424, -58169868]\) \(614716917569296417/19093020912\) \(78205013655552\) \([2]\) \(49152\) \(1.7625\)  
7728.o3 7728r4 \([0, 1, 0, -81184, 8073716]\) \(14447092394873377/1439452851984\) \(5895998881726464\) \([2, 4]\) \(49152\) \(1.7625\)  
7728.o1 7728r5 \([0, 1, 0, -1266144, 547941492]\) \(54804145548726848737/637608031452\) \(2611642496827392\) \([4]\) \(98304\) \(2.1091\)  
7728.o6 7728r6 \([0, 1, 0, 100256, 39208820]\) \(27207619911317663/177609314617308\) \(-727487752672493568\) \([4]\) \(98304\) \(2.1091\)