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Properties

Label 43706.f
Number of curves $3$
Conductor $43706$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 43706.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
43706.f1 43706b3 \([1, 1, 0, -772454, -261632876]\) \(-10730978619193/6656\) \(-31616693828096\) \([]\) \(423360\) \(1.9112\)  
43706.f2 43706b2 \([1, 1, 0, -7599, -511379]\) \(-10218313/17576\) \(-83487832139816\) \([]\) \(141120\) \(1.3619\)  
43706.f3 43706b1 \([1, 1, 0, 806, 14774]\) \(12167/26\) \(-123502710266\) \([]\) \(47040\) \(0.81256\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 43706.f have rank \(1\).

Complex multiplication

The elliptic curves in class 43706.f do not have complex multiplication.

Modular form 43706.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 3 q^{5} + q^{6} + q^{7} - q^{8} - 2 q^{9} + 3 q^{10} - 6 q^{11} - q^{12} - q^{13} - q^{14} + 3 q^{15} + q^{16} + 3 q^{17} + 2 q^{18} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.