Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -792, -17172]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -792, -17172]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -792, -17172]) 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 21294o have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 21294o do not have complex multiplication.

Modular form 21294.2.a.o

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);
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} + q^{14} + q^{16} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labeled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 21294o

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
21294.q5 21294o1 \([1, -1, 0, -792, -17172]\) \(-15625/28\) \(-98524825308\) \([2]\) \(17280\) \(0.79970\) \(\Gamma_0(N)\)-optimal
21294.q4 21294o2 \([1, -1, 0, -16002, -774630]\) \(128787625/98\) \(344836888578\) \([2]\) \(34560\) \(1.1463\)  
21294.q6 21294o3 \([1, -1, 0, 6813, 361557]\) \(9938375/21952\) \(-77243463041472\) \([2]\) \(51840\) \(1.3490\)  
21294.q3 21294o4 \([1, -1, 0, -54027, 3975453]\) \(4956477625/941192\) \(3311813477903112\) \([2]\) \(103680\) \(1.6956\)  
21294.q2 21294o5 \([1, -1, 0, -259362, 51051924]\) \(-548347731625/1835008\) \(-6456922951385088\) \([2]\) \(155520\) \(1.8983\)  
21294.q1 21294o6 \([1, -1, 0, -4153122, 3258731412]\) \(2251439055699625/25088\) \(88278243475968\) \([2]\) \(311040\) \(2.2449\)