Properties

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

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 43681.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
43681.a1 43681m3 \([0, 1, 1, -341599980, -2430217915988]\) \(-52893159101157376/11\) \(-916791127892651\) \([]\) \(4320000\) \(3.1679\)  
43681.a2 43681m2 \([0, 1, 1, -451370, -211562048]\) \(-122023936/161051\) \(-13422738903476303291\) \([]\) \(864000\) \(2.3632\)  
43681.a3 43681m1 \([0, 1, 1, -14560, 1601232]\) \(-4096/11\) \(-916791127892651\) \([]\) \(172800\) \(1.5584\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 43681.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{10} + 2 q^{12} + 4 q^{13} - 4 q^{14} + q^{15} - 4 q^{16} + 2 q^{17} + 4 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.