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Properties

Label 216849.q
Number of curves $4$
Conductor $216849$
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, 1, 0, -411039, 100014102]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -411039, 100014102]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -411039, 100014102]) 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 216849.q have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(41\)\(1\)
\(43\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 216849.q do not have complex multiplication.

Modular form 216849.2.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);
 
\(q + q^{2} - q^{3} - q^{4} + 2 q^{5} - q^{6} - 3 q^{8} + q^{9} + 2 q^{10} + q^{12} + 2 q^{13} - 2 q^{15} - q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 216849.q

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
216849.q1 216849q3 \([1, 1, 0, -411039, 100014102]\) \(1616855892553/22851963\) \(108549206361475083\) \([2]\) \(2112000\) \(2.0736\)  
216849.q2 216849q2 \([1, 1, 0, -49624, -1832645]\) \(2845178713/1347921\) \(6402765258632961\) \([2, 2]\) \(1056000\) \(1.7270\)  
216849.q3 216849q1 \([1, 1, 0, -41219, -3236280]\) \(1630532233/1161\) \(5514871023801\) \([2]\) \(528000\) \(1.3805\) \(\Gamma_0(N)\)-optimal
216849.q4 216849q4 \([1, 1, 0, 177311, -13678652]\) \(129784785047/92307627\) \(-438470850489346107\) \([2]\) \(2112000\) \(2.0736\)