Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -1173057, 489313853]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -1173057, 489313853]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -1173057, 489313853]) 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 19602.i have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 19602.i do not have complex multiplication.

Modular form 19602.2.a.i

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);
 
\(q - q^{2} + q^{4} - 2 q^{7} - q^{8} - 2 q^{13} + 2 q^{14} + q^{16} + 3 q^{17} + q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 19602.i

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
19602.i1 19602d4 \([1, -1, 0, -1173057, 489313853]\) \(-189613868625/128\) \(-120509459123328\) \([]\) \(170100\) \(2.0176\)  
19602.i2 19602d3 \([1, -1, 0, -11457, 961725]\) \(-1159088625/2097152\) \(-300933848236032\) \([]\) \(56700\) \(1.4683\)  
19602.i3 19602d1 \([1, -1, 0, -567, -5307]\) \(-140625/8\) \(-1147971528\) \([]\) \(8100\) \(0.49536\) \(\Gamma_0(N)\)-optimal
19602.i4 19602d2 \([1, -1, 0, 3063, -10873]\) \(3375/2\) \(-1882960298802\) \([]\) \(24300\) \(1.0447\)