↑

Properties

Label 13872b
Number of curves $6$
Conductor $13872$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, 193, -1338]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, 193, -1338]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, 193, -1338]) 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 13872b have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 3 T + 5 T^{2}\) 1.5.d
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + T + 11 T^{2}\) 1.11.b
\(13\) \( 1 - 3 T + 13 T^{2}\) 1.13.ad
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 7 T + 23 T^{2}\) 1.23.ah
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 13872b do not have complex multiplication.

Modular form 13872.2.a.b

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);
 
\(q - q^{3} + 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 13872b

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
13872.p5 13872b1 \([0, -1, 0, 193, -1338]\) \(2048/3\) \(-1158603312\) \([2]\) \(5120\) \(0.42468\) \(\Gamma_0(N)\)-optimal
13872.p4 13872b2 \([0, -1, 0, -1252, -12320]\) \(35152/9\) \(55612958976\) \([2, 2]\) \(10240\) \(0.77125\)  
13872.p2 13872b3 \([0, -1, 0, -18592, -969488]\) \(28756228/3\) \(74150611968\) \([2]\) \(20480\) \(1.1178\)  
13872.p3 13872b4 \([0, -1, 0, -7032, 218880]\) \(1556068/81\) \(2002066523136\) \([2, 2]\) \(20480\) \(1.1178\)  
13872.p1 13872b5 \([0, -1, 0, -111072, 14285088]\) \(3065617154/9\) \(444903671808\) \([2]\) \(40960\) \(1.4644\)  
13872.p6 13872b6 \([0, -1, 0, 4528, 856992]\) \(207646/6561\) \(-324334776748032\) \([2]\) \(40960\) \(1.4644\)