Properties

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

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 1638q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.s3 1638q1 \([1, -1, 1, -509, 4421]\) \(19968681097/628992\) \(458535168\) \([4]\) \(768\) \(0.43680\) \(\Gamma_0(N)\)-optimal
1638.s2 1638q2 \([1, -1, 1, -1229, -10267]\) \(281397674377/96589584\) \(70413806736\) \([2, 2]\) \(1536\) \(0.78338\)  
1638.s1 1638q3 \([1, -1, 1, -17609, -894787]\) \(828279937799497/193444524\) \(141021057996\) \([2]\) \(3072\) \(1.1299\)  
1638.s4 1638q4 \([1, -1, 1, 3631, -74419]\) \(7264187703863/7406095788\) \(-5399043829452\) \([2]\) \(3072\) \(1.1299\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1638.2.a.q

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 2 q^{5} - q^{7} + q^{8} + 2 q^{10} + 4 q^{11} + q^{13} - q^{14} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.