↑

Properties

Label 352800bv
Number of curves $4$
Conductor $352800$
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, 0, 0, -223436325, 1282024068500]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -223436325, 1282024068500]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -223436325, 1282024068500]) 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 352800bv have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 352800bv do not have complex multiplication.

Modular form 352800.2.a.bv

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);
 
\(q - 4 q^{11} - 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 352800bv

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
352800.bv3 352800bv1 \([0, 0, 0, -223436325, 1282024068500]\) \(14383655824793536/45209390625\) \(3877434066680015625000000\) \([2, 2]\) \(70778880\) \(3.5853\) \(\Gamma_0(N)\)-optimal
352800.bv1 352800bv2 \([0, 0, 0, -3572280075, 82180042537250]\) \(7347751505995469192/72930375\) \(50039642934603000000000\) \([2]\) \(141557760\) \(3.9319\)  
352800.bv4 352800bv3 \([0, 0, 0, -129668700, 2366352884000]\) \(-43927191786304/415283203125\) \(-2279502684703125000000000000\) \([2]\) \(141557760\) \(3.9319\)  
352800.bv2 352800bv4 \([0, 0, 0, -319905075, 67193099750]\) \(5276930158229192/3050936350875\) \(2093335809859549647000000000\) \([2]\) \(141557760\) \(3.9319\)