Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("x1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3648.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.x1 3648r3 \([0, 1, 0, -5603329, -5107121569]\) \(74220219816682217473/16416\) \(4303355904\) \([2]\) \(46080\) \(2.1412\)  
3648.x2 3648r2 \([0, 1, 0, -350209, -79885729]\) \(18120364883707393/269485056\) \(70643890520064\) \([2, 2]\) \(23040\) \(1.7946\)  
3648.x3 3648r4 \([0, 1, 0, -339969, -84766113]\) \(-16576888679672833/2216253521952\) \(-580977563258585088\) \([4]\) \(46080\) \(2.1412\)  
3648.x4 3648r1 \([0, 1, 0, -22529, -1176993]\) \(4824238966273/537919488\) \(141012366262272\) \([2]\) \(11520\) \(1.4480\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3648.2.a.x

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} - 6q^{17} + q^{19} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.