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Properties

Label 39039.u
Number of curves $2$
Conductor $39039$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 39039.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
39039.u1 39039s2 \([1, 0, 1, -1096, -2455]\) \(66184391125/37202781\) \(81734509857\) \([2]\) \(25344\) \(0.78402\)  
39039.u2 39039s1 \([1, 0, 1, 269, -271]\) \(985074875/586971\) \(-1289575287\) \([2]\) \(12672\) \(0.43745\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 39039.u have rank \(1\).

Complex multiplication

The elliptic curves in class 39039.u do not have complex multiplication.

Modular form 39039.2.a.u

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} - q^{7} - 3 q^{8} + q^{9} - q^{11} - q^{12} - q^{14} - q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.