↑

Properties

Label 265335z
Number of curves $2$
Conductor $265335$
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, 1, 82549, 18423257]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 1, 82549, 18423257]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 1, 82549, 18423257]) 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 265335z have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T^{2}\) 1.2.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 265335z do not have complex multiplication.

Modular form 265335.2.a.z

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);
 
\(q - q^{3} - 2 q^{4} - q^{5} + q^{9} + 2 q^{12} + q^{13} + q^{15} + 4 q^{16} + 6 q^{17} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

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 265335z

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
265335.z2 265335z1 \([0, -1, 1, 82549, 18423257]\) \(229376/675\) \(-183066845738409675\) \([]\) \(2268000\) \(1.9958\) \(\Gamma_0(N)\)-optimal
265335.z1 265335z2 \([0, -1, 1, -3632141, 2671083386]\) \(-19539165184/46875\) \(-12712975398500671875\) \([]\) \(6804000\) \(2.5451\)