Properties

Label 3648bg
Number of curves $4$
Conductor $3648$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3648bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.w3 3648bg1 \([0, 1, 0, -689, 6735]\) \(2211014608/513\) \(8404992\) \([2]\) \(768\) \(0.31963\) \(\Gamma_0(N)\)-optimal
3648.w2 3648bg2 \([0, 1, 0, -769, 4991]\) \(768400132/263169\) \(17247043584\) \([2, 2]\) \(1536\) \(0.66620\)  
3648.w1 3648bg3 \([0, 1, 0, -5089, -137569]\) \(111223479026/3518667\) \(461198721024\) \([2]\) \(3072\) \(1.0128\)  
3648.w4 3648bg4 \([0, 1, 0, 2271, 37215]\) \(9878111854/10097379\) \(-1323483660288\) \([4]\) \(3072\) \(1.0128\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3648bg have rank \(1\).

Complex multiplication

The elliptic curves in class 3648bg do not have complex multiplication.

Modular form 3648.2.a.bg

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} - 2 q^{13} - 2 q^{15} + 2 q^{17} - q^{19} + O(q^{20})\)  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.