Properties

Label 38640.r
Number of curves $4$
Conductor $38640$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 38640.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.r1 38640bo4 \([0, -1, 0, -29496, -1810704]\) \(692895692874169/51420783750\) \(210619530240000\) \([2]\) \(147456\) \(1.4942\)  
38640.r2 38640bo2 \([0, -1, 0, -5976, 146160]\) \(5763259856089/1143116100\) \(4682203545600\) \([2, 2]\) \(73728\) \(1.1476\)  
38640.r3 38640bo1 \([0, -1, 0, -5656, 165616]\) \(4886171981209/270480\) \(1107886080\) \([2]\) \(36864\) \(0.80100\) \(\Gamma_0(N)\)-optimal
38640.r4 38640bo3 \([0, -1, 0, 12424, 852720]\) \(51774168853511/107398242630\) \(-439903201812480\) \([2]\) \(147456\) \(1.4942\)  

Rank

sage: E.rank()
 

The elliptic curves in class 38640.r have rank \(1\).

Complex multiplication

The elliptic curves in class 38640.r do not have complex multiplication.

Modular form 38640.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\)  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.