Properties

Label 31939.e
Number of curves $3$
Conductor $31939$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 31939.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31939.e1 31939b3 \([0, -1, 1, -1293249, -565640702]\) \(-50357871050752/19\) \(-90251980579\) \([]\) \(207360\) \(1.8902\)  
31939.e2 31939b2 \([0, -1, 1, -15689, -799487]\) \(-89915392/6859\) \(-32580964989019\) \([]\) \(69120\) \(1.3409\)  
31939.e3 31939b1 \([0, -1, 1, 1121, -1012]\) \(32768/19\) \(-90251980579\) \([]\) \(23040\) \(0.79161\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 31939.e have rank \(1\).

Complex multiplication

The elliptic curves in class 31939.e do not have complex multiplication.

Modular form 31939.2.a.e

sage: E.q_eigenform(10)
 
\(q + 2q^{3} - 2q^{4} + 3q^{5} + q^{7} + q^{9} - 3q^{11} - 4q^{12} + 4q^{13} + 6q^{15} + 4q^{16} + 3q^{17} - q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.