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Properties

Label 19166a
Number of curves $6$
Conductor $19166$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -713, 14773]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -713, 14773]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -713, 14773]) 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 19166a have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(7\)\(1 - T\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 19166a do not have complex multiplication.

Modular form 19166.2.a.a

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);
 
\(q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} + q^{7} + q^{8} + q^{9} - 2 q^{12} + 4 q^{13} + q^{14} + q^{16} - 6 q^{17} + q^{18} - 2 q^{19} + O(q^{20})\) 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 labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 19166a

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
19166.a5 19166a1 \([1, 0, 0, -713, 14773]\) \(-15625/28\) \(-71840339452\) \([2]\) \(17280\) \(0.77337\) \(\Gamma_0(N)\)-optimal
19166.a4 19166a2 \([1, 0, 0, -14403, 663679]\) \(128787625/98\) \(251441188082\) \([2]\) \(34560\) \(1.1199\)  
19166.a6 19166a3 \([1, 0, 0, 6132, -309680]\) \(9938375/21952\) \(-56322826130368\) \([2]\) \(51840\) \(1.3227\)  
19166.a3 19166a4 \([1, 0, 0, -48628, -3387192]\) \(4956477625/941192\) \(2414841170339528\) \([2]\) \(103680\) \(1.6693\)  
19166.a2 19166a5 \([1, 0, 0, -233443, -43557759]\) \(-548347731625/1835008\) \(-4708128486326272\) \([2]\) \(155520\) \(1.8720\)  
19166.a1 19166a6 \([1, 0, 0, -3738083, -2782083455]\) \(2251439055699625/25088\) \(64368944148992\) \([2]\) \(311040\) \(2.2186\)