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Properties

Label 24986.b
Number of curves $3$
Conductor $24986$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 24986.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
24986.b1 24986b3 \([1, 1, 0, -441599, 112767301]\) \(-10730978619193/6656\) \(-5907224500736\) \([]\) \(181440\) \(1.7714\)  
24986.b2 24986b2 \([1, 1, 0, -4344, 217864]\) \(-10218313/17576\) \(-15598764697256\) \([]\) \(60480\) \(1.2221\)  
24986.b3 24986b1 \([1, 1, 0, 461, -6049]\) \(12167/26\) \(-23075095706\) \([]\) \(20160\) \(0.67277\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 24986.b have rank \(2\).

Complex multiplication

The elliptic curves in class 24986.b do not have complex multiplication.

Modular form 24986.2.a.b

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.