Properties

Label 39984p
Number of curves $4$
Conductor $39984$
CM no
Rank $1$
Graph

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

Elliptic curves in class 39984p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
39984.dm3 39984p1 \([0, 1, 0, -5847, 170148]\) \(11745974272/357\) \(672011088\) \([2]\) \(30720\) \(0.79168\) \(\Gamma_0(N)\)-optimal
39984.dm2 39984p2 \([0, 1, 0, -6092, 154860]\) \(830321872/127449\) \(3838527334656\) \([2, 2]\) \(61440\) \(1.1383\)  
39984.dm4 39984p3 \([0, 1, 0, 10568, 867908]\) \(1083360092/3306177\) \(-398303659901952\) \([2]\) \(122880\) \(1.4848\)  
39984.dm1 39984p4 \([0, 1, 0, -26672, -1532700]\) \(17418812548/1753941\) \(211301790422016\) \([2]\) \(122880\) \(1.4848\)  

Rank

sage: E.rank()
 

The elliptic curves in class 39984p have rank \(1\).

Complex multiplication

The elliptic curves in class 39984p do not have complex multiplication.

Modular form 39984.2.a.p

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