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Properties

Label 16562bo
Number of curves $6$
Conductor $16562$
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, 1, -4313, -222461]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -4313, -222461]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -4313, -222461]) 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 16562bo have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(7\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(11\) \( 1 + 5 T + 11 T^{2}\) 1.11.f
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - T + 29 T^{2}\) 1.29.ab
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 16562bo do not have complex multiplication.

Modular form 16562.2.a.bo

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);
 
\(q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{8} + q^{9} + 2 q^{12} + q^{16} - 6 q^{17} + q^{18} + 2 q^{19} + O(q^{20})\) 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 16562bo

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
16562.bv5 16562bo1 \([1, 1, 1, -4313, -222461]\) \(-15625/28\) \(-15900339057148\) \([2]\) \(34560\) \(1.2233\) \(\Gamma_0(N)\)-optimal
16562.bv4 16562bo2 \([1, 1, 1, -87123, -9927793]\) \(128787625/98\) \(55651186700018\) \([2]\) \(69120\) \(1.5699\)  
16562.bv6 16562bo3 \([1, 1, 1, 37092, 4630205]\) \(9938375/21952\) \(-12465865820804032\) \([2]\) \(103680\) \(1.7727\)  
16562.bv3 16562bo4 \([1, 1, 1, -294148, 50208829]\) \(4956477625/941192\) \(534473997066972872\) \([2]\) \(207360\) \(2.1192\)  
16562.bv2 16562bo5 \([1, 1, 1, -1412083, 647136433]\) \(-548347731625/1835008\) \(-1042044620449251328\) \([2]\) \(311040\) \(2.3220\)  
16562.bv1 16562bo6 \([1, 1, 1, -22611443, 41375346865]\) \(2251439055699625/25088\) \(14246703795204608\) \([2]\) \(622080\) \(2.6685\)