Properties

Label 32448bi
Number of curves $4$
Conductor $32448$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("bi1")
 
E.isogeny_class()
 

Elliptic curves in class 32448bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32448.cg4 32448bi1 \([0, 1, 0, -15604, 101882]\) \(1360251712/771147\) \(238219473915072\) \([2]\) \(129024\) \(1.4486\) \(\Gamma_0(N)\)-optimal
32448.cg2 32448bi2 \([0, 1, 0, -158409, -24203529]\) \(22235451328/123201\) \(2435758881214464\) \([2, 2]\) \(258048\) \(1.7952\)  
32448.cg3 32448bi3 \([0, 1, 0, -70529, -50831169]\) \(-245314376/6908733\) \(-1092718907326365696\) \([4]\) \(516096\) \(2.1418\)  
32448.cg1 32448bi4 \([0, 1, 0, -2531169, -1550837313]\) \(11339065490696/351\) \(55515871936512\) \([2]\) \(516096\) \(2.1418\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32448bi have rank \(0\).

Complex multiplication

The elliptic curves in class 32448bi do not have complex multiplication.

Modular form 32448.2.a.bi

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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.

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

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.