Properties

Label 1638.p
Number of curves $4$
Conductor $1638$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1638.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.p1 1638m3 \([1, -1, 1, -8615, -305585]\) \(3592121380875/246064\) \(4843277712\) \([2]\) \(1728\) \(0.91259\)  
1638.p2 1638m4 \([1, -1, 1, -8075, -345977]\) \(-2958077788875/946054564\) \(-18621191983212\) \([2]\) \(3456\) \(1.2592\)  
1638.p3 1638m1 \([1, -1, 1, -215, 623]\) \(40530337875/18264064\) \(493129728\) \([6]\) \(576\) \(0.36329\) \(\Gamma_0(N)\)-optimal
1638.p4 1638m2 \([1, -1, 1, 745, 4079]\) \(1695802078125/1272491584\) \(-34357272768\) \([6]\) \(1152\) \(0.70986\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1638.p have rank \(0\).

Complex multiplication

The elliptic curves in class 1638.p do not have complex multiplication.

Modular form 1638.2.a.p

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{7} + q^{8} + q^{13} + q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.