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Properties

Label 156816.bw
Number of curves $4$
Conductor $156816$
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([0, 0, 0, -18768915, -31297317678]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -18768915, -31297317678]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -18768915, -31297317678]) 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 156816.bw have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(11\)\(1\)
 
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.ac
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 156816.bw do not have complex multiplication.

Modular form 156816.2.a.bw

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);
 
\(q + 2 q^{7} - 2 q^{13} + 3 q^{17} - q^{19} + O(q^{20})\) 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 labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 156816.bw

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
156816.bw1 156816bk4 \([0, 0, 0, -18768915, -31297317678]\) \(-189613868625/128\) \(-493606744569151488\) \([]\) \(4082400\) \(2.7108\)  
156816.bw2 156816bk3 \([0, 0, 0, -183315, -61367086]\) \(-1159088625/2097152\) \(-1232625042374787072\) \([]\) \(1360800\) \(2.1615\)  
156816.bw3 156816bk1 \([0, 0, 0, -9075, 348722]\) \(-140625/8\) \(-4702091378688\) \([]\) \(194400\) \(1.1885\) \(\Gamma_0(N)\)-optimal
156816.bw4 156816bk2 \([0, 0, 0, 49005, 646866]\) \(3375/2\) \(-7712605383892992\) \([]\) \(583200\) \(1.7378\)