Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3648.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3648.i1 3648b3 \([0, -1, 0, -27393, -1735935]\) \(8671983378625/82308\) \(21576548352\) \([2]\) \(6912\) \(1.1452\)  
3648.i2 3648b4 \([0, -1, 0, -26753, -1821567]\) \(-8078253774625/846825858\) \(-221990317719552\) \([2]\) \(13824\) \(1.4918\)  
3648.i3 3648b1 \([0, -1, 0, -513, 513]\) \(57066625/32832\) \(8606711808\) \([2]\) \(2304\) \(0.59592\) \(\Gamma_0(N)\)-optimal
3648.i4 3648b2 \([0, -1, 0, 2047, 2049]\) \(3616805375/2105352\) \(-551905394688\) \([2]\) \(4608\) \(0.94249\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3648.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{3} - 4 q^{7} + q^{9} + 4 q^{13} + 6 q^{17} - q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.