↑

Properties

Label 1650q
Number of curves $4$
Conductor $1650$
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([1, 0, 0, -34838, -2505708]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -34838, -2505708]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -34838, -2505708]) 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 1650q have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 + T + 13 T^{2}\) 1.13.b
\(17\) \( 1 - 8 T + 17 T^{2}\) 1.17.ai
\(19\) \( 1 + 5 T + 19 T^{2}\) 1.19.f
\(23\) \( 1 - 9 T + 23 T^{2}\) 1.23.aj
\(29\) \( 1 - 5 T + 29 T^{2}\) 1.29.af
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1650q do not have complex multiplication.

Modular form 1650.2.a.q

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);
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + q^{11} + q^{12} - 2 q^{13} + q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + O(q^{20})\) 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 1650q

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
1650.r3 1650q1 \([1, 0, 0, -34838, -2505708]\) \(299270638153369/1069200\) \(16706250000\) \([2]\) \(3840\) \(1.1796\) \(\Gamma_0(N)\)-optimal
1650.r2 1650q2 \([1, 0, 0, -35338, -2430208]\) \(312341975961049/17862322500\) \(279098789062500\) \([2, 2]\) \(7680\) \(1.5261\)  
1650.r1 1650q3 \([1, 0, 0, -104088, 9876042]\) \(7981893677157049/1917731420550\) \(29964553446093750\) \([2]\) \(15360\) \(1.8727\)  
1650.r4 1650q4 \([1, 0, 0, 25412, -9902458]\) \(116149984977671/2779502343750\) \(-43429724121093750\) \([2]\) \(15360\) \(1.8727\)