Properties

Label 43681.i
Number of curves $3$
Conductor $43681$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 43681.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
43681.i1 43681j3 \([0, -1, 1, -33605249, -74971104968]\) \(-50357871050752/19\) \(-1583548311814579\) \([]\) \(1458000\) \(2.7046\)  
43681.i2 43681j2 \([0, -1, 1, -407689, -106457473]\) \(-89915392/6859\) \(-571660940565063019\) \([]\) \(486000\) \(2.1553\)  
43681.i3 43681j1 \([0, -1, 1, 29121, -94238]\) \(32768/19\) \(-1583548311814579\) \([]\) \(162000\) \(1.6060\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 43681.i have rank \(1\).

Complex multiplication

The elliptic curves in class 43681.i do not have complex multiplication.

Modular form 43681.2.a.i

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